#6630918

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629016
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35278	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(18,24)(20,26)(21,22)(25,28)(27,32)(30,31);(2,5)(6,8)(17,18,21,24,23,26,22,20)(19,27,30,32,29,25,31,28);(10,13)(14,16)(17,19,23,29)(18,25,26,27)(20,32,24,28)(21,30,22,31);(1,3,4,7)(2,6,5,8)(17,20,21,18,23,24,22,26)(19,28,30,27,29,32,31,25);(9,11,12,15)(10,14,13,16)(19,29)(25,27)(28,32)(30,31)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,c^b=ci,d^b=di,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538114);