#6480615

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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	#25729	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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)(4,5)(7,8)(11,12)(14,16)(17,18,24,27)(19,22,26,23)(20,29,21,25)(28,31,30,32);(9,10)(11,16)(12,14)(13,15)(17,19,23,18,24,26,22,27)(20,30,31,29,21,28,32,25);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20)(18,25)(19,28)(21,24)(22,32)(23,31)(26,30)(27,29);(10,15)(14,16)(17,21,24,20)(18,25,27,29)(19,28,26,30)(22,31,23,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(17,22,24,23)(18,19,27,26)(20,31,21,32)(25,30,29,28)

Polycyclic presentation:

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

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, 6387811);