#6969874

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#28143	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1417 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)(10,14)(11,12)(17,18,21,27,22,19,23,25)(20,26,30,32,29,24,28,31);(1,2,6,3)(4,7,5,8)(9,10,13,14)(11,16,12,15)(18,19)(24,26)(25,27)(31,32);(1,3,6,2)(4,7,5,8)(9,11,13,12)(10,15,14,16)(17,19,22,18)(20,26,29,24)(21,27,23,25)(28,32,30,31);(1,4,6,5)(2,7,3,8)(9,12,13,11)(10,15,14,16)(17,20,22,29)(18,24,19,26)(21,30,23,28)(25,32,27,31);(1,5,6,4)(2,7,3,8)(17,21,22,23)(18,25,19,27)(20,28,29,30)(24,31,26,32)

Polycyclic presentation:

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

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, 6877070);