#7596432

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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1455 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,26)(19,27,30,44)(20,31,35,47)(21,36,34,25)(23,37,39,28)(24,40,42,32)(29,41,33,38)(43,48,46,45);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(19,28)(20,32)(21,34)(23,27)(24,31)(25,36)(29,33)(30,37)(35,40)(38,41)(39,44)(42,47);(10,13)(14,15)(17,19,21,28,22,30,34,37)(18,23,25,27,26,39,36,44)(20,33,32,48,35,29,40,45)(24,41,31,46,42,38,47,43);(9,11,12,16)(10,14,13,15)(17,20,34,40,22,35,21,32)(18,24,36,47,26,42,25,31)(19,29,37,48,30,33,28,45)(23,38,44,46,39,41,27,43);(1,3,4,8)(2,6,5,7)(19,30)(23,39)(27,44)(28,37)(29,33)(38,41)(43,46)(45,48)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gi,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, 7503628);