#7598515

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

1395 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)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,34)(21,40)(22,30)(23,29)(24,28)(25,32)(26,37)(31,47)(33,42)(35,44)(36,41)(38,46)(39,45)(43,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,32)(18,25,29,27)(20,35,38,36)(21,24,33,22)(26,43,45,47)(28,40,30,42)(31,39,48,37)(34,41,46,44);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,22,37,23,38,24,39)(18,26,28,34,29,45,30,46)(19,31,33,35,32,48,21,36)(25,41,42,43,27,44,40,47);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,24,32,23,33,22,19)(18,27,30,42,29,25,28,40)(20,36,39,48,38,35,37,31)(26,44,46,43,45,41,34,47);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,28,29,30)(19,21,32,33)(20,37,38,39)(25,40,27,42)(26,34,45,46)(31,36,48,35)(41,47,44,43)

Polycyclic presentation:

2^9|b^2=i,c^2=fh,d^2=fg,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,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7505711);