#6464207

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2066 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,25)(19,28,20,26)(21,27,22,31)(24,30,32,29);(9,11)(10,13)(12,15)(14,16)(17,19,23,20)(18,24,25,32)(21,29,22,30)(26,31,28,27);(2,5)(6,8)(10,14)(11,15)(18,25)(24,32)(26,28)(27,31);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,23,19)(18,26,25,28)(21,29,22,30)(24,31,32,27);(17,21,23,22)(18,27,25,31)(19,29,20,30)(24,26,32,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gh,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371403);