#508295

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
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	C₂⁵	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

788 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415491);