#496947

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

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=cei,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, 404143);