#373542

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,22)(21,24)(25,26,30,33,29,32,34,27)(28,36,39,35,38,31,40,37);(1,3,5,4)(2,6,7,8)(10,13)(14,15)(17,19)(18,21)(20,24)(22,23)(25,27,34,32,29,33,30,26)(28,37,40,31,38,35,39,36);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(18,22)(19,24)(25,28,29,38)(26,31,32,36)(27,35,33,37)(30,39,34,40);(2,7)(6,8)(9,11,12,16)(10,14,13,15)(18,22)(21,23)(26,32)(27,33)(28,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=ef,d^2=fg,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280738);