#504662

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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

584 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=ei,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411858);