#6376564

order	512
nilpotency class	3
isoclinism	isoclinic class of #60771
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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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)(24,36)(25,30)(26,40)(27,31)(29,37)(32,46)(33,42)(34,43)(35,50)(38,44)(39,45)(41,47)(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)(22,28)(24,32)(25,33)(26,35)(27,34)(29,41)(30,42)(31,43)(36,46)(37,47)(38,48)(39,49)(40,50)(44,51)(45,52);(7,14)(10,13)(15,20)(17,19)(21,24,26,37,27,38,25,39)(22,29,30,36,31,45,40,44)(23,32,35,47,34,48,33,49)(28,41,42,46,43,52,50,51);(6,11)(10,13)(12,18)(17,19)(21,25,27,26)(22,30,31,40)(23,33,34,35)(24,37,38,39)(28,42,43,50)(29,44,45,36)(32,47,48,49)(41,51,52,46);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(23,34)(24,38)(28,43)(29,45)(33,35)(36,44)(37,39)(42,50)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^b=ei,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, 6283760);