#7595244

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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(19,24,32,25)(20,33,31,41)(21,36,39,45)(22,27,34,28)(26,43,46,38)(30,44,47,48)(35,40,37,42);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,32)(18,24,29,25)(20,22,31,34)(21,35,39,37)(26,44,46,48)(27,41,28,33)(30,43,47,38)(36,40,45,42);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,34,19,23,31,22,32)(18,25,27,41,29,24,28,33)(21,37,43,47,39,35,38,30)(26,40,45,48,46,42,36,44);(9,11,12,16)(10,14,13,15)(17,21,34,43,23,39,22,38)(18,26,27,45,29,46,28,36)(19,30,31,37,32,47,20,35)(24,40,33,48,25,42,41,44);(2,5)(6,8)(17,22,23,34)(18,27,29,28)(19,31,32,20)(21,38,39,43)(24,41,25,33)(26,45,46,36)(30,37,47,35)(40,44,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,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=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, 7502440);