#7597748

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1371 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fhi,d^2=gi,e^2=fi,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=cg,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, 7504944);