#7535498

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	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

1408 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)(12,14)(13,16)(17,18,22,27)(19,24,32,42)(20,34,37,25)(21,26,23,28)(29,44,35,39)(30,47,48,40)(31,41,33,43)(36,46,38,45);(2,6)(7,8)(11,16)(13,15)(17,19,22,32)(18,24,27,42)(20,35,37,29)(21,33,23,31)(25,44,34,39)(26,43,28,41)(30,38,48,36)(40,46,47,45);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,38)(22,37)(23,36)(24,39)(26,46)(27,34)(28,45)(30,33)(31,48)(32,35)(40,43)(41,47)(42,44);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,40)(26,45)(27,34)(28,46)(29,33)(31,35)(32,48)(39,43)(41,44)(42,47);(1,4,5,3)(2,7,6,8)(17,21,22,23)(18,26,27,28)(19,31,32,33)(20,36,37,38)(24,41,42,43)(25,45,34,46)(29,30,35,48)(39,40,44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=f,e^2=fi,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, 7442694);