#7013244

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#30503	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1407 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:

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

Polycyclic presentation:

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

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, 6920440);