#7301531

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	#42908	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1916 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7208727);