#506514

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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	#25777	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

821 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413710);