#510048

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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 ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

501 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417244);