#7603568

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18)(19,24)(20,34)(21,39)(22,27)(23,29)(25,37)(26,32)(28,31)(30,41)(33,44)(35,40)(36,48)(38,42)(43,45)(46,47);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19,23,21)(18,24,29,39)(20,35,38,46)(22,32,31,33)(25,41,43,48)(26,28,44,27)(30,45,36,37)(34,40,42,47);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,31,45,23,38,22,37)(18,25,27,42,29,43,28,34)(19,30,32,46,21,36,33,35)(24,40,44,48,39,47,26,41);(1,4,5,3)(2,6,8,7)(11,16)(14,15)(17,21,22,32,23,19,31,33)(18,26,28,39,29,44,27,24)(20,36,37,46,38,30,45,35)(25,41,34,47,43,48,42,40);(2,8)(6,7)(17,22,23,31)(18,27,29,28)(19,32,21,33)(20,37,38,45)(24,26,39,44)(25,42,43,34)(30,46,36,35)(40,41,47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,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=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, 7510764);