#6506504

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2338 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)(17,18,24,29)(19,30,20,32)(21,26,25,31)(22,27,23,28);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,31,24,20,22,26)(18,25,27,32,29,21,28,30);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,31,24,19,23,26)(18,25,28,30,29,21,27,32);(11,16)(13,15)(17,21,24,25)(18,26,29,31)(19,28,20,27)(22,32,23,30);(10,14)(13,15)(17,22,24,23)(18,27,29,28)(19,26,20,31)(21,30,25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,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, 6413700);