#6461219

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1974 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6368415);