#7611035

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1428 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)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,20,25)(21,26,36,44)(22,38,24,27)(30,45,32,39)(31,40,33,41)(34,47,46,42)(35,48,37,43);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,20)(18,25,28,29)(21,34,36,46)(22,31,24,33)(26,42,44,47)(27,40,38,41)(30,35,32,37)(39,48,45,43);(10,14)(11,15)(17,20,23,19)(18,25,28,29)(21,32,36,30)(22,31,24,33)(26,39,44,45)(27,41,38,40)(34,37,46,35)(42,48,47,43);(10,14)(13,16)(17,21,22,35,23,36,24,37)(18,26,38,48,28,44,27,43)(19,30,33,46,20,32,31,34)(25,39,40,47,29,45,41,42);(1,3,4,8)(2,6,5,7)(17,22,23,24)(18,27,28,38)(19,31,20,33)(21,35,36,37)(25,40,29,41)(26,43,44,48)(30,34,32,46)(39,47,45,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,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, 7518231);