#6697311

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1614 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)(10,14)(11,12)(17,18,22,23)(19,24,29,26)(20,25,21,30)(27,28,31,32);(9,10)(11,15)(12,16)(13,14)(17,19,22,29)(18,24,23,26)(20,27,21,31)(25,28,30,32);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(18,23)(19,26)(20,21)(24,29)(27,28)(31,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,20)(18,25)(19,27)(21,22)(23,30)(24,28)(26,32)(29,31);(10,14)(15,16)(17,21,22,20)(18,25,23,30)(19,28,29,32)(24,27,26,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,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, 6604507);