#6545512

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1637 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)(3,8)(4,6)(5,7)(17,18,24,28,23,27,22,26)(19,29,21,20,30,25,32,31);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,26)(21,24)(22,32)(23,30)(27,29)(28,31);(10,13)(14,16)(17,20)(18,21)(19,28)(22,25)(23,31)(24,29)(26,30)(27,32);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,32)(24,30)(25,28)(26,29)(27,31);(11,15)(14,16)(17,22,23,24)(18,26,27,28)(19,21,30,32)(20,25,31,29)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452708);