#6762140

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=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, 6669336);