#506512

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	C₂²×#135	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

792 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)(13,14,20,36,34,15,21,37,19,25,18,24,17,23,39,27)(16,28,43,26,30,44,35,22,33,42,31,40,32,29,38,41);(5,7,8,12)(6,9,10,11)(13,15,18,27,34,25,39,36,19,23,20,37,17,14,21,24)(16,29,31,22,30,28,38,40,33,44,43,41,32,42,35,26);(2,4)(13,16)(14,22)(15,26)(17,30)(18,35)(19,33)(20,38)(21,31)(23,40)(24,29)(25,41)(27,42)(28,37)(32,34)(36,44)(39,43);(6,10)(9,11)(13,17,19,34)(14,23,25,15)(16,30,33,32)(18,21,20,39)(22,40,41,26)(24,37,36,27)(28,44,42,29)(31,38,43,35)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,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, 413708);