#6376558

order	512
nilpotency class	3
isoclinism	isoclinic class of #60753
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

1716 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)(23,28)(24,35)(25,30)(26,40)(27,31)(29,36)(32,47)(33,42)(34,44)(37,45)(38,43)(39,46)(41,50)(48,51)(49,52);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23,27,34)(22,28,31,44)(24,32,37,49)(25,38,26,33)(29,41,46,51)(30,43,40,42)(35,47,45,52)(36,50,39,48);(7,14)(10,13)(15,20)(17,19)(21,24,25,39,27,37,26,36)(22,29,40,45,31,46,30,35)(23,32,38,48,34,49,33,50)(28,41,42,52,44,51,43,47);(6,11)(10,13)(12,18)(17,19)(21,25,27,26)(22,30,31,40)(23,33,34,38)(24,36,37,39)(28,42,44,43)(29,45,46,35)(32,48,49,50)(41,47,51,52);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(24,37)(29,46)(32,49)(35,45)(36,39)(41,51)(47,52)(48,50)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283754);