#6802725

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1534 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)(17,18,23,26,22,27,21,28)(19,29,31,39,32,24,33,41)(20,34,35,25,36,42,37,40)(30,44,47,43,46,48,45,38);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,22,32)(18,24,27,29)(20,30,36,46)(21,33,23,31)(25,38,40,43)(26,41,28,39)(34,48,42,44)(35,47,37,45);(3,8)(5,7)(10,13)(14,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,34)(27,40)(28,42)(29,43)(31,45)(32,46)(33,47)(39,44)(41,48);(2,6)(5,7)(17,21,22,23)(18,26,27,28)(19,31,32,33)(20,35,36,37)(24,39,29,41)(25,34,40,42)(30,45,46,47)(38,44,43,48);(9,11,12,16)(10,14,13,15)(18,27)(19,32)(25,40)(26,28)(30,46)(31,33)(34,42)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6709921);