#7608636

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1638 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)(17,18,24,30)(19,25,20,26)(21,35,38,27)(22,28,23,29)(31,42,33,40)(32,41,34,43)(36,48,47,44)(37,46,39,45);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,24,20)(18,25,30,26)(21,36,38,47)(22,32,23,34)(27,44,35,48)(28,41,29,43)(31,39,33,37)(40,46,42,45);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,24,19)(18,26,30,25)(21,33,38,31)(22,32,23,34)(27,42,35,40)(28,41,29,43)(36,37,47,39)(44,45,48,46);(10,14)(13,16)(17,21,22,39,24,38,23,37)(18,27,28,46,30,35,29,45)(19,31,34,47,20,33,32,36)(25,40,43,48,26,42,41,44);(2,5)(6,8)(17,22,24,23)(18,28,30,29)(19,32,20,34)(21,37,38,39)(25,41,26,43)(27,45,35,46)(31,47,33,36)(40,48,42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=hi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7515832);