#6612422

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	#37782	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1812 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)(11,14)(12,15)(13,16)(18,22)(19,25)(20,27)(23,30)(24,29)(28,31);(9,11)(10,13)(12,16)(14,15)(17,18,20,22,21,24,27,29)(19,23,26,28,25,31,32,30);(2,5)(6,8)(11,16)(13,14)(17,19)(18,23)(20,26)(21,25)(22,28)(24,31)(27,32)(29,30);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15)(18,24)(22,29)(23,31)(28,30);(11,16)(13,14)(19,25)(23,31)(26,32)(28,30)

Polycyclic presentation:

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

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, 6519618);