#508226

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

736 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^b=di,d^c=de,f^c=fh,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, 415422);