#507164

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

604 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)(13,14,21,28)(15,23,33,44)(16,34,31,22)(17,36,38,43)(18,39,30,26)(19,40,29,25)(20,41,35,27)(24,32,42,37);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15,29,37,20,17,30,16,21,33,19,32,35,38,18,31)(14,22,39,43,27,42,40,44,28,34,26,36,41,24,25,23);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,17,35,37,19,15,21,31,30,38,20,32,29,33)(14,23,25,24,41,36,26,34,28,44,40,42,27,43,39,22);(6,11)(10,12)(13,17,19,31,20,33,18,37,21,38,29,16,35,15,30,32)(14,24,26,44,27,22,25,36,28,42,39,23,41,34,40,43)

Polycyclic presentation:

2^9|a^2=i,b^2=egh,c^2=e,d^2=f,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg

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, 414360);