#6805572

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(17,18,23,27)(19,28,31,38)(20,25,33,42)(21,26,22,36)(24,34,39,30)(29,43,46,48)(32,40,35,41)(37,44,47,45);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14);(3,8)(5,7)(10,13)(14,15)(17,19,21,34,23,31,22,30)(18,24,36,38,27,39,26,28)(20,29,35,44,33,46,32,45)(25,37,40,48,42,47,41,43);(2,6)(5,7)(17,20)(18,25)(19,29)(21,35)(22,32)(23,33)(24,37)(26,41)(27,42)(28,43)(30,45)(31,46)(34,44)(36,40)(38,48)(39,47);(9,11,12,16)(10,14,13,15)(17,21,23,22)(18,26,27,36)(19,30,31,34)(20,32,33,35)(24,38,39,28)(25,40,42,41)(29,44,46,45)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=gi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6712768);