#6701982

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#25174	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1417 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,21,24)(19,23,28,32)(20,25,30,22)(26,27,29,31);(9,11,14,10)(12,15,13,16)(17,19,21,28)(18,23,24,32)(20,27,30,31)(22,26,25,29);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(18,24)(19,26)(22,25)(23,31)(27,32)(28,29);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(17,20)(18,25)(19,27)(21,30)(22,24)(23,29)(26,32)(28,31);(19,26)(20,30)(22,25)(23,27)(28,29)(31,32)

Polycyclic presentation:

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

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, 6609178);