#7183851

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

1651 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,d^2=fi,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, 7091047);