#6840300

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#30454	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1437 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)(9,10,13,15)(11,16,12,14)(17,18,23,22)(19,21,26,31)(20,24,30,29)(25,27,32,28);(9,10)(11,14)(12,16)(13,15)(17,19)(18,21)(20,27)(22,31)(23,26)(24,32)(25,29)(28,30);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,18,24,23,30,22,29)(19,25,31,28,26,32,21,27);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(17,18,23,22)(19,21,26,31)(20,24,30,29)(25,27,32,28);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,18,19,23,31,22,26)(20,28,24,32,30,27,29,25)

Polycyclic presentation:

2^9|a^2=fi,c^2=fhi,d^2=gi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6747496);