#6477064

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
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	#24382	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1887 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,31,32)(20,26,24,25)(21,27,22,30);(9,11)(10,13)(12,15)(14,16)(17,19,22,26,23,31,21,25)(18,24,27,29,28,20,30,32);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20,23,24)(18,25,28,26)(19,30,31,27)(21,29,22,32);(11,15)(13,16)(17,20)(18,26)(19,30)(21,29)(22,32)(23,24)(25,28)(27,31);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,21,23,22)(18,27,28,30)(19,25,31,26)(20,32,24,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384260);