#508451

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

508 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=e,d^2=e,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, 415647);