#6494961

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=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, 6402157);