#503534

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

796 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=efh,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=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, 410730);