#7144116

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1591 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,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, 7051312);