#6411101

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=df

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, 6318297);