#508469

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

489 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=e,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415665);