#381673

order	512
nilpotency class	3
isoclinism	isoclinic class of #219508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

765 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)(3,5)(4,6)(7,8)(9,10)(11,21,25,36)(12,17)(13,18)(14,19)(15,20)(16,24,35,30)(22,23,26,38)(27,28)(29,31)(32,37)(33,39,40,34);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,26)(13,23)(14,30)(15,25)(17,33)(18,34)(19,36)(20,35)(21,37)(22,27)(24,32)(28,40)(29,38)(31,39);(2,6)(5,8)(9,12,15,27)(10,17,20,28)(11,22,25,26)(13,19,29,37)(14,31,32,18)(16,33,35,40)(21,23,36,38)(24,39,30,34);(9,13,14,28,15,29,32,17)(10,18,19,27,20,31,37,12)(11,23,30,40,25,38,24,33)(16,34,36,22,35,39,21,26)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288869);