#7595243

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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

1544 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)(17,18,24,29)(19,25,32,42)(20,33,36,46)(21,39,31,41)(22,28,23,27)(26,38,45,37)(30,47,48,43)(34,44,35,40);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,32)(18,25,29,42)(20,34,36,35)(21,23,31,22)(26,43,45,47)(27,41,28,39)(30,37,48,38)(33,44,46,40);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,22,37,24,36,23,38)(18,26,27,33,29,45,28,46)(19,30,31,35,32,48,21,34)(25,40,39,47,42,44,41,43);(9,11,12,16)(10,14,13,15)(17,21,23,32,24,31,22,19)(18,25,28,41,29,42,27,39)(20,35,38,30,36,34,37,48)(26,44,46,47,45,40,33,43);(2,5)(6,8)(20,36)(26,45)(30,48)(33,46)(34,35)(37,38)(40,44)(43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,d^2=gi,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,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, 7502439);