#6458927

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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

1724 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)(4,7)(5,8)(9,10,12,14)(11,13,15,16)(18,23)(19,27)(21,28)(24,32)(25,31)(26,30);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,18,22,26)(19,24,27,31)(20,25,29,32)(21,30,28,23);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,25)(26,31)(30,32);(1,4,5,3)(2,7,8,6)(11,15)(13,16)(17,20,22,29)(18,25,26,32)(19,28,27,21)(23,31,30,24);(19,27)(20,29)(24,31)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366123);