#506508

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	C₂×#2215	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

824 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)(13,14,20,34,33,38,31,40,17,23,18,24,19,25,36,44)(15,26,35,37,16,30,43,39,27,21,28,22,29,42,32,41);(5,7,8,12)(6,9,10,11)(13,15,20,35,33,16,31,43,17,27,18,28,19,29,36,32)(14,21,34,22,38,42,40,41,23,26,24,37,25,30,44,39);(2,4)(13,16,17,29)(14,22,23,37)(15,19,27,33)(18,28,20,35)(21,24,26,34)(25,41,38,39)(30,44,42,40)(31,32,36,43);(6,10)(9,11)(14,23)(21,26)(22,37)(24,34)(25,38)(30,42)(39,41)(40,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,c^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, 413704);