#7582829

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1274 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)(11,16)(12,14)(13,15)(17,18,23,27)(19,24,31,39)(20,25,33,42)(21,34,22,26)(28,36,45,48)(29,37,30,38)(32,41,35,40)(43,46,44,47);(2,6)(7,8)(9,10)(11,16)(12,14)(13,15)(17,19,23,31)(18,24,27,39)(20,28,33,45)(21,30,22,29)(25,36,42,48)(26,37,34,38)(32,43,35,44)(40,47,41,46);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(17,20)(18,25)(19,28)(21,35)(22,32)(23,33)(24,36)(26,41)(27,42)(29,43)(30,44)(31,45)(34,40)(37,46)(38,47)(39,48);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(19,29)(21,22)(24,37)(26,34)(28,43)(30,31)(32,35)(36,46)(38,39)(40,41)(44,45)(47,48);(1,4,5,3)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,21,23,22)(18,26,27,34)(19,30,31,29)(20,32,33,35)(24,37,39,38)(25,40,42,41)(28,43,45,44)(36,47,48,46)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7490025);