#6729665

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=g,e^2=gi,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,h^c=hi,h^e=hi

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, 6636861);