#498529

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405725);