#508455

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=e,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415651);