#501461

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

544 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)(7,12)(8,10)(9,11)(13,14,21,36,17,24,35,38,19,26,18,25,20,27,33,41)(15,22,32,44,31,43,37,42,16,30,29,40,28,39,34,23);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,22)(16,19)(17,31)(18,29)(20,28)(21,32)(23,41)(24,43)(25,40)(26,30)(27,39)(33,34)(35,37)(36,44)(38,42);(2,4)(5,8,9,7)(6,10,11,12)(13,16,19,15)(14,23,26,42)(17,31,20,28)(18,34,21,37)(22,38,30,41)(24,40,27,44)(25,43,36,39)(29,35,32,33);(6,11)(10,12)(13,17,19,20)(14,24,26,27)(15,28,16,31)(18,33,21,35)(22,39,30,43)(23,40,42,44)(25,41,36,38)(29,37,32,34)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408657);