#510011

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

590 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=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, 417207);