#7248019

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	#42674	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7155215);