#6546614

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	#37670	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1623 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=ehi

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, 6453810);