#496945

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	#392	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

612 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)(7,11)(13,14,18,24,19,22,32,38,17,23,20,33,16,29,31,41)(15,21,28,36,26,37,40,44,27,25,34,39,30,35,42,43);(5,6,8,9)(7,10,11,12)(13,14,20,33,19,22,31,41,17,23,18,24,16,29,32,38)(15,25,34,36,26,35,42,44,27,21,28,39,30,37,40,43);(2,4)(6,9)(10,12)(13,15)(14,21)(16,30)(17,27)(18,28)(19,26)(20,34)(22,37)(23,25)(24,36)(29,35)(31,42)(32,40)(33,39)(38,44)(41,43);(5,7,8,11)(6,10,9,12)(13,16,17,19)(14,22,23,29)(15,26,27,30)(18,31,20,32)(21,35,25,37)(24,38,33,41)(28,40,34,42)(36,43,39,44)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=di,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, 404141);