#207139

order	512
nilpotency class	3
isoclinism	isoclinic class of #162118
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

874 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,21)(13,15)(14,20)(16,18)(19,23)(22,24)(25,26,30,34,29,32,28,33)(27,31,37,40,36,38,35,39);(2,5)(6,7)(9,11,18,24,13,19,12,20)(10,14,21,23,15,22,16,17)(25,27,28,37,29,36,30,35)(26,31,34,39,32,38,33,40);(1,3,4,8)(2,6,5,7)(10,15)(14,22)(16,21)(17,23)(26,32)(31,38)(33,34)(39,40);(11,19)(14,22)(17,23)(20,24)(25,28,29,30)(26,33,32,34)(27,35,36,37)(31,39,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fh,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114335);