#6456199

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	#35236	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1730 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,31)(29,32);(2,6)(7,8)(19,28)(20,30)(22,29)(23,24)(26,32)(27,31);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,21,30)(18,23,25,27)(20,29,28,22)(24,26,31,32);(1,4,5,3)(2,8,6,7)(17,20)(18,24)(19,29)(21,28)(22,30)(23,26)(25,31)(27,32);(9,11,12,15)(10,14,13,16)(18,25)(23,27)(24,31)(26,32)

Polycyclic presentation:

2^9|c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

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, 6363395);