#7149947

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	#42664	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,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, 7057143);