#6999068

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1380 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)(17,18,21,26)(19,28,30,23)(20,25,33,44)(22,36,35,27)(24,37,42,34)(29,43,32,39)(31,40,47,45)(38,46,41,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,30)(18,23,26,28)(20,32,33,29)(22,31,35,47)(24,41,42,38)(25,43,44,39)(27,40,36,45)(34,48,37,46);(9,11)(10,13)(12,16)(14,15)(17,20,21,33)(18,24,26,42)(19,29,30,32)(22,37,35,34)(23,38,28,41)(25,27,44,36)(31,48,47,46)(39,40,43,45);(3,8)(6,7)(11,16)(13,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,39)(24,36)(26,44)(27,42)(28,43)(30,32)(31,46)(35,37)(38,45)(40,41)(47,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,26)(19,30)(20,33)(27,36)(31,47)(34,37)(38,41)(39,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=fi,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,e^c=ei,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, 6906264);