#7597387

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	#25765	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1646 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)(19,28)(20,32)(21,36)(22,37)(23,27)(24,31)(25,34)(26,38)(29,44)(30,40)(33,43)(35,48)(39,46)(41,47)(42,45);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,31)(18,24,27,28)(20,33,35,46)(21,22,30,38)(25,41,42,44)(26,40,37,36)(29,45,47,34)(32,39,48,43);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,22,34,23,35,38,45)(18,25,26,32,27,42,37,48)(19,29,21,33,31,47,30,46)(24,39,36,41,28,43,40,44);(9,11,12,16)(10,14,13,15)(17,21,22,31,23,30,38,19)(18,24,26,36,27,28,37,40)(20,33,34,47,35,46,45,29)(25,39,32,41,42,43,48,44);(1,4,5,3)(2,7,6,8)(17,22,23,38)(18,26,27,37)(19,30,31,21)(20,34,35,45)(24,40,28,36)(25,32,42,48)(29,46,47,33)(39,44,43,41)

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=g,e^2=f,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, 7504583);