#401004

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1638 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1638 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,12,32,11,16,10,17)(7,13,30,41,20,36,21,35)(8,22,31,14,24,42,25,38)(9,26,29,40,27,39,28,15)(18,43,51,33,46,52,45,61)(19,47,49,62,50,58,48,34)(23,54,56,59,57,37,55,53)(44,64,67,68,65,60,66,63);(1,2)(3,4)(5,7,10,30,11,20,12,21)(6,13,32,35,16,36,17,41)(8,18,31,45,24,46,25,51)(9,19,29,48,27,50,28,49)(14,33,22,52,38,61,42,43)(15,34,39,47,40,62,26,58)(23,44,55,67,57,65,56,66)(37,60,53,68,54,64,59,63);(2,4)(5,8)(6,14)(7,18)(9,23)(10,31)(11,24)(12,25)(13,33)(15,37)(16,38)(17,42)(19,44)(20,46)(21,51)(22,32)(26,59)(27,57)(28,56)(29,55)(30,45)(34,60)(35,52)(36,61)(39,53)(40,54)(41,43)(47,68)(48,67)(49,66)(50,65)(58,63)(62,64);(5,9,10,29,11,27,12,28)(6,15,32,39,16,40,17,26)(7,19,30,48,20,50,21,49)(8,23,31,55,24,57,25,56)(13,34,35,47,36,62,41,58)(14,37,22,53,38,54,42,59)(18,44,45,67,46,65,51,66)(33,60,52,68,61,64,43,63)

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308200);