#7605188

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 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)(10,14)(11,12)(17,18,21,23)(19,27,31,39)(20,32,34,42)(22,35,29,26)(24,38,36,28)(25,41,37,33)(30,45,46,40)(43,47,48,44);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(18,23)(19,28)(20,33)(22,29)(24,39)(25,42)(27,36)(30,46)(31,38)(32,37)(34,41)(43,48);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,19,29,38,21,31,22,28)(18,24,35,39,23,36,26,27)(20,30,41,47,34,46,33,44)(25,40,42,48,37,45,32,43);(9,11,13,12)(10,16,14,15)(17,20,29,41,21,34,22,33)(18,25,35,42,23,37,26,32)(19,30,38,47,31,46,28,44)(24,40,39,48,36,45,27,43);(1,4,5,3)(2,7,6,8)(9,12,13,11)(10,15,14,16)(18,23)(19,31)(25,37)(26,35)(28,38)(30,46)(32,42)(44,47)

Polycyclic presentation:

2^9|a^2=fghi,c^2=fh,d^2=gi,e^2=fi,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, 7512384);