#502734

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	C₂×#3341	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

880 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 166 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)(7,11)(8,9)(10,12)(13,14,18,22)(15,19,26,34)(16,20,29,37)(17,21,28,36)(23,31,40,43)(24,32,39,42)(25,33,30,38)(27,35,41,44);(6,9)(10,11)(13,15)(14,19)(16,23)(17,30)(18,26)(20,31)(21,38)(22,34)(24,27)(25,28)(29,40)(32,35)(33,36)(37,43)(39,41)(42,44);(1,2,3,4)(15,23)(16,27)(17,28)(19,31)(20,35)(21,36)(24,25)(26,40)(29,41)(30,39)(32,33)(34,43)(37,44)(38,42);(5,7,8,12)(6,10,9,11)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,g^c=gh,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, 409930);