#6971059

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907945
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	#24248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=fi,e^2=fhi,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=dfh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6878255);