#6872952

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

1522 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fi,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, 6780148);