#7598359

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	C₂×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1428 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)(19,29,32,41)(20,34,38,42)(21,27,22,26)(24,33,40,30)(25,35,44,37)(31,43,36,39)(45,47,48,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(19,30)(20,35)(21,22)(24,29)(25,42)(26,27)(32,33)(34,44)(37,38)(40,41)(45,48)(46,47);(10,13)(14,15)(17,19,22,33,23,32,21,30)(18,24,27,41,28,40,26,29)(20,36,35,46,38,31,37,47)(25,43,42,45,44,39,34,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,22,35,23,38,21,37)(18,25,27,42,28,44,26,34)(19,31,33,47,32,36,30,46)(24,39,41,48,40,43,29,45);(1,4,5,3)(2,7,6,8)(19,32)(24,40)(29,41)(30,33)(31,36)(39,43)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=fi,c^2=h,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,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, 7505555);