#6464159

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1769 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,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, 6371355);