#498249

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

552 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)(7,10)(8,9)(11,12)(13,14,19,25,20,15,26,28,18,24,33,39,17,23,21,34)(16,22,31,37,29,27,40,41,30,36,42,44,32,38,35,43);(6,9)(7,11)(13,15,19,28,20,24,26,39,18,23,33,34,17,14,21,25)(16,27,31,41,29,36,40,44,30,38,42,43,32,22,35,37);(2,4)(13,16)(14,22)(15,27)(17,32)(18,30)(19,31)(20,29)(21,35)(23,38)(24,36)(25,37)(26,40)(28,41)(33,42)(34,43)(39,44);(5,7,8,11)(6,10,9,12)(13,17,18,20)(14,23,24,15)(16,29,30,32)(19,21,33,26)(22,27,36,38)(25,34,39,28)(31,40,42,35)(37,41,44,43)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=dei,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, 405445);