#502858

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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	#33179	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 889 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fhi,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di

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, 410054);