#7182604

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(2,4)(8,16)(11,17)(15,20)(18,19)(21,22,27,29,25,31,26,32)(23,33,39,41,34,28,36,43)(24,30,40,44,37,46,38,45)(35,47,50,51,48,42,49,52);(1,2,3,4)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23)(22,28)(24,35)(25,34)(26,36)(27,39)(29,43)(30,42)(31,33)(32,41)(37,48)(38,49)(40,50)(44,52)(45,51)(46,47);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(22,29)(23,34)(24,37)(26,27)(28,41)(30,45)(31,32)(33,43)(42,52)(44,46)(47,51)(49,50);(2,4)(7,14)(10,13)(15,20)(18,19)(21,24,25,37)(22,30,31,46)(23,35,34,48)(26,38,27,40)(28,42,33,47)(29,44,32,45)(36,49,39,50)(41,51,43,52);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(23,34)(24,37)(28,33)(30,46)(36,39)(38,40)(41,43)(44,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,d^2=i,e^2=hi,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, 7089800);