#498187

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

580 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)(6,9)(10,12)(13,14,18,23,19,15,24,27,17,22,32,39,33,26,20,34)(16,21,30,38,31,25,36,41,29,37,40,44,43,28,35,42);(5,6)(7,10)(8,9)(11,12)(13,15,18,27,19,22,24,39,17,26,32,34,33,14,20,23)(16,28,30,42,31,21,36,38,29,25,40,41,43,37,35,44);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,29)(14,21,22,37)(15,25,26,28)(18,30,32,40)(19,31,33,43)(20,35,24,36)(23,38,39,44)(27,41,34,42);(6,9)(10,12)(14,22)(15,26)(16,29)(23,39)(27,34)(30,40)(31,43)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405383);