#509802

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24778	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 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)(7,11)(8,10)(9,12)(13,14,20,36,29,41,28,40,19,25,18,24,15,21,37,44)(16,30,33,39,26,43,34,22,17,32,31,38,27,42,35,23);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,29)(14,21,25,41)(16,26,17,27)(18,28,20,37)(22,38,23,39)(24,40,36,44)(30,43,32,42)(31,35,33,34);(2,4)(13,16,19,17)(14,22,25,23)(15,26,29,27)(18,34,20,35)(21,38,41,39)(24,42,36,43)(28,31,37,33)(30,44,32,40);(6,10)(7,12)(13,17,19,16)(14,23,25,22)(15,27,29,26)(18,35,20,34)(21,39,41,38)(24,43,36,42)(28,33,37,31)(30,40,32,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=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, 416998);