#7598331

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=fgi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,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, 7505527);