#7172363

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1264 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)(10,13)(11,15)(17,18,23,25,22,27,21,28)(19,24,35,37,31,39,30,40)(20,26,36,41,33,44,34,43)(29,38,46,47,32,42,45,48);(1,2,5,6)(3,7,4,8)(10,13)(11,15)(17,19,22,31)(18,24,27,39)(20,32,33,29)(21,30,23,35)(25,37,28,40)(26,42,44,38)(34,46,36,45)(41,48,43,47);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(18,25)(21,23)(24,37)(26,41)(27,28)(30,35)(34,36)(38,47)(39,40)(42,48)(43,44)(45,46);(1,4,5,3)(2,7,6,8)(10,13)(14,16)(17,20)(18,26)(19,29)(21,34)(22,33)(23,36)(24,38)(25,41)(27,44)(28,43)(30,45)(31,32)(35,46)(37,47)(39,42)(40,48);(9,11,12,15)(10,14,13,16)(18,27)(20,33)(24,39)(25,28)(29,32)(34,36)(37,40)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=f,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7079559);