#504447

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

639 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=i,d^2=fi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411643);