#401002

order	512
nilpotency class	3
isoclinism	isoclinic class of #395181
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

1662 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1662 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)(7,18)(8,23)(9,27,29,41)(10,30)(11,17)(12,16)(13,21)(14,26)(15,28,42,31)(19,45)(20,35,52,61)(22,37)(24,55,58,62)(25,39)(32,36)(33,40)(34,49)(38,56,54,57)(43,48)(44,51,60,50)(46,59,66,68)(47,53)(63,67,64,65);(1,2)(3,4)(5,7,10,21,11,22,12,32)(6,13,16,18,17,36,30,37)(8,19,25,47,26,48,33,49)(9,20,31,51,29,52,28,50)(14,34,39,43,23,53,40,45)(15,35,27,44,42,61,41,60)(24,46,57,67,58,66,56,65)(38,59,55,63,54,68,62,64);(2,4)(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,33)(13,34)(15,38)(16,39)(17,23)(18,43)(20,46)(21,47)(22,48)(27,55)(28,56)(29,58)(30,40)(31,57)(32,49)(35,59)(36,53)(37,45)(41,62)(42,54)(44,63)(50,65)(51,67)(52,66)(60,64)(61,68);(5,9,12,28,11,29,10,31)(6,15,30,41,17,42,16,27)(7,20,32,50,22,52,21,51)(8,24,33,56,26,58,25,57)(13,35,37,60,36,61,18,44)(14,38,40,62,23,54,39,55)(19,46,49,65,48,66,47,67)(34,59,45,64,53,68,43,63)

Polycyclic presentation:

2^9|a^2=fi,b^2=efi,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^b=hi,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, 308198);