#388481

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295677);