#7572620

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fg,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7479816);