#6536742

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1484 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)(4,8)(5,6)(17,18,24,28)(19,22,27,23)(20,30,31,25)(21,26,32,29);(9,10)(11,14)(12,13)(15,16)(17,19,23,18,24,27,22,28)(20,26,21,30,31,29,32,25);(2,6)(7,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,23)(22,32)(24,31)(26,27)(28,30);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,26)(19,25)(20,22)(23,31)(24,32)(27,30)(28,29);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,22,24,23)(18,27,28,19)(20,21,31,32)(25,29,30,26)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ei,d^c=di,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, 6443938);