#7602540

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1361 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)(9,10,12,14)(11,15,16,13)(17,18,23,28)(19,30,33,25)(20,34,32,42)(21,36,39,45)(22,29,24,27)(26,40,44,38)(31,43,47,48)(35,46,37,41);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,24)(21,37)(22,32)(23,33)(26,43)(27,42)(28,30)(29,34)(31,40)(35,39)(36,46)(38,47)(41,45)(44,48);(1,3,4,7)(2,6,5,8)(17,20,22,33,23,32,24,19)(18,25,27,34,28,30,29,42)(21,35,38,47,39,37,40,31)(26,41,36,48,44,46,45,43);(11,16)(13,15)(17,21,24,40,23,39,22,38)(18,26,29,45,28,44,27,36)(19,31,32,37,33,47,20,35)(25,41,42,43,30,46,34,48);(2,5)(6,8)(10,14)(13,15)(17,22,23,24)(18,27,28,29)(19,32,33,20)(21,38,39,40)(25,42,30,34)(26,36,44,45)(31,37,47,35)(41,43,46,48)

Polycyclic presentation:

2^9|a^2=ghi,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=cgi,d^b=dh,g^b=gi,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, 7509736);