#207217

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

804 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)(9,10,15,12)(11,18,22,24)(13,17,14,23)(16,20,19,21)(25,26,29,32,28,31,35,37)(27,30,34,39,33,38,36,40);(2,5)(6,7)(9,11,13,20,15,22,14,21)(10,16,23,24,12,19,17,18)(25,27,29,36,28,33,35,34)(26,30,37,39,31,38,32,40);(1,3,4,8)(2,6,5,7)(9,12,15,10)(11,19,22,16)(13,17,14,23)(18,21,24,20)(26,31)(30,38)(32,37)(39,40);(9,13,15,14)(10,17,12,23)(11,20,22,21)(16,18,19,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114413);