#6971382

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
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	#24238	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 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)(4,8)(5,7)(10,13)(11,15)(17,18,22,26)(19,28,30,24)(20,29,31,32)(21,27,23,25);(1,2,6,3)(4,7,5,8)(9,10,12,13)(11,14,15,16)(17,19,22,30)(18,24,26,28)(20,27,31,25)(21,32,23,29);(1,3,6,2)(4,7,5,8)(10,13)(14,16)(17,20,21,28,22,31,23,24)(18,19,25,32,26,30,27,29);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,26,27)(19,29,30,32)(20,24,31,28);(1,5,6,4)(2,7,3,8)

Polycyclic presentation:

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

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, 6878578);