#508416

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×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 578 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)(13,14,16,19)(15,20,24,31)(17,21,26,32)(18,22,27,33)(23,35,41,34)(25,36,30,40)(28,39,43,37)(29,38,42,44);(6,9)(10,12)(14,19)(15,23)(17,28)(18,27)(20,34)(21,37)(24,41)(25,42)(26,43)(29,30)(31,35)(32,39)(36,38)(40,44);(2,4)(7,11)(10,12)(13,15)(14,20)(16,24)(17,29)(18,30)(19,31)(21,38)(22,40)(23,28)(25,27)(26,42)(32,44)(33,36)(34,37)(35,39)(41,43);(5,7,8,11)(6,10,9,12)(14,19)(15,24)(21,32)(22,33)(23,41)(25,30)(29,42)(37,39)

Polycyclic presentation:

2^9|a^2=ef,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, 415612);