#7594828

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

1569 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)(17,18,24,30)(19,25,21,40)(20,34,39,44)(22,28,23,29)(26,37,45,38)(27,32,42,33)(31,43,36,48)(35,41,46,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,21)(18,25,30,40)(20,35,39,46)(22,32,23,33)(26,43,45,48)(27,28,42,29)(31,37,36,38)(34,41,44,47);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,23,38,24,39,22,37)(18,26,28,34,30,45,29,44)(19,31,32,35,21,36,33,46)(25,41,27,43,40,47,42,48);(9,11,12,16)(10,14,13,15)(17,21,22,32,24,19,23,33)(18,27,29,25,30,42,28,40)(20,36,37,35,39,31,38,46)(26,43,44,41,45,48,34,47);(2,5)(6,8)(17,22,24,23)(18,28,30,29)(19,32,21,33)(20,37,39,38)(25,42,40,27)(26,34,45,44)(31,35,36,46)(41,48,47,43)

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=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7502024);