#6871522

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1692 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)(10,13)(11,15)(17,18,22,24,20,25,21,26)(19,27,30,31,28,23,29,32);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,20,28)(18,23,25,27)(21,29,22,30)(24,32,26,31);(10,13)(14,16)(18,24)(19,28)(21,22)(23,31)(25,26)(27,32);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(19,28)(23,27)(29,30)(31,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=hi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6778718);