#6795063

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(3,7)(4,6)(5,8)(17,18,23,26,22,27,21,25)(19,20,28,30,24,32,29,31);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,13)(14,16)(17,19,22,24)(18,20,27,32)(21,29,23,28)(25,31,26,30);(3,8)(5,7)(10,13)(14,15)(17,20)(18,24)(19,27)(21,30)(22,32)(23,31)(25,29)(26,28);(2,6)(5,7)(17,21,22,23)(18,25,27,26)(19,28,24,29)(20,30,32,31);(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,26,27,25)(19,29,24,28)(20,30,32,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6702259);