#7610574

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1445 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,30)(19,25,32,40)(20,26,34,42)(21,27,37,45)(22,28,23,29)(31,39,47,48)(33,41,35,43)(36,44,38,46);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,22)(21,33)(23,34)(24,32)(26,28)(27,41)(29,42)(30,40)(31,36)(35,37)(38,47)(39,44)(43,45)(46,48);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20)(18,26)(19,22)(21,35)(23,32)(24,34)(25,28)(27,43)(29,40)(30,42)(31,36)(33,37)(38,47)(39,44)(41,45)(46,48);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21,23,36,24,37,22,38)(18,27,29,44,30,45,28,46)(19,31,20,33,32,47,34,35)(25,39,26,41,40,48,42,43);(2,8)(6,7)(17,22,24,23)(18,28,30,29)(19,20,32,34)(21,36,37,38)(25,26,40,42)(27,44,45,46)(31,35,47,33)(39,43,48,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fh,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7517770);