#380753

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 298 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) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287949);