#498215

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

560 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=dei,f^c=fh,h^c=hi

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, 405411);