#6471126

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	#29600	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1815 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)(19,25)(20,26)(21,23)(22,30)(24,27)(28,32)(29,31);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(18,23)(24,31)(25,32)(26,30);(11,15)(13,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,31)(25,30)(26,32);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,19)(18,25)(20,29)(21,28)(22,27)(23,32)(24,30)(26,31);(10,14)(13,16)(17,20,21,22)(18,26,23,30)(19,27,28,29)(24,32,31,25)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6378322);