#7575269

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1302 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7482465);