#498435

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

619 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)(6,10)(11,12)(13,14,18,24,19,25,31,39,17,23,20,32,16,22,30,38)(15,21,28,36,26,34,40,43,27,35,33,42,29,37,41,44);(5,6,9,10)(7,11,8,12)(13,14,20,32,19,25,30,38,17,23,18,24,16,22,31,39)(15,21,33,42,26,34,41,44,27,35,28,36,29,37,40,43);(2,4)(5,7,9,8)(6,11,10,12)(13,15,17,27)(14,21,23,35)(16,29,19,26)(18,28,20,33)(22,37,25,34)(24,36,32,42)(30,41,31,40)(38,44,39,43);(5,8,9,7)(6,11,10,12)(13,16,17,19)(14,22,23,25)(15,26,27,29)(18,30,20,31)(21,34,35,37)(24,38,32,39)(28,40,33,41)(36,43,42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,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, 405631);