#6469323

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	#25731	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1968 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18,22,25)(19,26,27,32)(20,24,21,30)(23,28,31,29);(9,10)(11,16)(12,14)(13,15)(17,19,21,29,22,27,20,28)(18,23,24,26,25,31,30,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,30)(21,24)(22,25)(26,29)(27,31)(28,32);(10,14)(13,16)(19,27)(23,31)(26,32)(28,29);(1,4,5,3)(2,6,7,8)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,26,31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,e^2=gh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=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, 6376519);