#499650

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 752 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=i,d^2=efh,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,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, 406846);