#395183

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

1712 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^c=hi,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, 302379);