#7147508

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,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, 7054704);