#7595908

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	#45332	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1782 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)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,23,29)(19,24,32,25)(20,33,31,40)(21,36,38,44)(22,27,34,28)(26,42,45,37)(30,43,47,48)(35,39,46,41);(2,6)(7,8)(10,13)(11,16)(17,19,23,32)(18,24,29,25)(20,22,31,34)(21,35,38,46)(26,43,45,48)(27,40,28,33)(30,42,47,37)(36,39,44,41);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,34,19,23,31,22,32)(18,25,27,40,29,24,28,33)(21,35,42,30,38,46,37,47)(26,41,44,43,45,39,36,48);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,21,22,37,23,38,34,42)(18,26,28,36,29,45,27,44)(19,30,20,35,32,47,31,46)(24,39,40,43,25,41,33,48);(2,6)(7,8)(10,13)(14,15)(17,22,23,34)(18,27,29,28)(19,31,32,20)(21,37,38,42)(24,40,25,33)(26,44,45,36)(30,46,47,35)(39,43,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,d^2=fg,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, 7503104);