#7594816

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	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1571 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,29)(19,31,21,25)(20,34,38,45)(22,30,24,28)(26,37,46,39)(27,33,42,40)(32,44,36,43)(35,47,48,41);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,35)(21,23)(22,33)(24,40)(26,43)(27,30)(28,42)(29,31)(32,39)(34,41)(36,37)(38,48)(44,46)(45,47);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,22,37,23,38,24,39)(18,26,28,45,29,46,30,34)(19,32,40,48,21,36,33,35)(25,41,27,44,31,47,42,43);(9,11,12,16)(10,14,13,15)(17,21,22,33,23,19,24,40)(18,27,28,31,29,42,30,25)(20,36,37,35,38,32,39,48)(26,44,45,47,46,43,34,41);(2,5)(6,8)(17,22,23,24)(18,28,29,30)(19,33,21,40)(20,37,38,39)(25,42,31,27)(26,45,46,34)(32,35,36,48)(41,43,47,44)

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=gi,e^2=i,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, 7502012);