#508918

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
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	C₂×#2218	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

742 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 162 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)(7,10)(8,9)(11,12)(13,14,18,24,16,30,31,40,17,23,32,39,19,22,33,21)(15,25,28,42,27,36,43,38,26,20,34,37,29,41,44,35);(1,2,3,4)(6,9)(10,12)(13,15)(14,20)(16,27)(17,26)(18,28)(19,29)(21,38)(22,36)(23,25)(24,37)(30,41)(31,43)(32,34)(33,44)(35,40)(39,42);(2,4)(5,7)(6,10)(8,11)(9,12)(14,21)(15,26)(16,19)(18,33)(20,35)(22,24)(23,40)(25,38)(28,43)(30,39)(31,32)(34,44)(36,42)(37,41);(7,11)(10,12)(13,16,17,19)(14,22,23,30)(15,27,26,29)(18,31,32,33)(20,36,25,41)(21,39,40,24)(28,43,34,44)(35,37,38,42)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416114);