#6376582

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1709 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)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22,26,30)(23,31,35,47)(24,28,37,44)(25,39,32,29)(27,40,42,34)(33,46,36,50)(38,52,48,45)(41,51,43,49);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19);(7,14)(10,13)(15,20)(17,19)(21,23,32,40,26,35,25,34)(22,27,39,47,30,42,29,31)(24,36,48,49,37,33,38,51)(28,43,52,46,44,41,45,50);(6,11)(10,13)(12,18)(17,19)(21,24)(22,28)(23,33)(25,38)(26,37)(27,41)(29,45)(30,44)(31,46)(32,48)(34,49)(35,36)(39,52)(40,51)(42,43)(47,50);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(24,37)(28,44)(33,36)(38,48)(41,43)(45,52)(46,50)(49,51)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^c=di,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283778);