#7608711

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(20,30)(24,39)(27,32)(31,46)(34,40)(36,41)(42,47)(44,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18,22,26)(19,23,28,37)(20,31,30,46)(21,35,33,25)(24,40,39,34)(27,42,32,47)(29,45,43,38)(36,48,41,44);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,19)(18,23)(20,32)(21,29)(22,28)(24,41)(25,38)(26,37)(27,30)(31,47)(33,43)(34,48)(35,45)(36,39)(40,44)(42,46);(10,14)(13,16)(17,20,33,40,22,30,21,34)(18,24,35,46,26,39,25,31)(19,27,43,48,28,32,29,44)(23,36,45,47,37,41,38,42);(2,5)(6,8)(19,28)(23,37)(27,32)(29,43)(36,41)(38,45)(42,47)(44,48)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei

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, 7515907);