#508418

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

554 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=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, 415614);