#6461264

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1940 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,22,31,21,25,20,26)(19,27,30,23,28,24,29,32);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,28)(18,23,25,32)(20,30,22,29)(24,31,27,26);(1,3,4,8)(2,6,5,7)(10,14)(11,15)(17,19)(18,24)(20,29)(21,28)(22,30)(23,26)(25,27)(31,32);(11,15)(13,16)(19,28)(23,32)(24,27)(29,30);(18,25)(23,32)(24,27)(26,31)

Polycyclic presentation:

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

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, 6368460);