#6536506

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	#37670	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1849 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)(17,18)(19,26)(20,25)(21,30)(22,24)(23,28)(27,31)(29,32);(9,10)(11,14)(12,13)(15,16)(17,19,21,28,22,27,20,29)(18,23,25,26,24,32,30,31);(2,5)(6,8)(10,13)(14,16)(18,24)(19,27)(25,30)(28,29);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,30)(21,25)(22,24)(26,28)(27,32)(29,31);(11,15)(14,16)(17,20,22,21)(18,25,24,30)(19,28,27,29)(23,31,32,26)

Polycyclic presentation:

2^9|b^2=f,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,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, 6443702);