#510026

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	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

587 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 136 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)(13,14,17,22,29,36,42,33,16,19,28,35,18,23,30,20)(15,24,26,39,32,44,43,34,25,40,41,37,27,38,31,21);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(14,19)(15,25)(20,33)(22,35)(23,36)(26,41)(27,32)(31,43);(2,4)(5,8,9,10)(6,11,12,7)(14,20)(17,30)(18,29)(19,33)(21,24)(22,23)(26,31)(27,32)(28,42)(34,40)(35,36)(37,44)(38,39)(41,43);(6,12)(7,11)(13,15,16,25)(14,21,19,34)(17,31,28,43)(18,32,29,27)(20,24,33,40)(22,38,35,44)(23,39,36,37)(26,42,41,30)

Polycyclic presentation:

2^9|a^2=efi,b^2=ei,c^2=ei,d^2=i,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, 417222);