#207145

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

856 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)(11,18)(12,17)(13,22)(14,16)(15,19)(20,24)(21,23)(25,26,28,32,29,31,36,37)(27,30,34,39,33,38,35,40);(2,5)(6,7)(9,11,12,21,14,20,13,19)(10,15,22,24,16,23,17,18)(25,27,28,35,29,33,36,34)(26,30,37,39,31,38,32,40);(1,3,4,8)(2,6,5,7)(10,16)(15,23)(17,22)(18,24);(9,12,14,13)(10,17,16,22)(11,19,20,21)(15,24,23,18)(27,33)(30,38)(34,35)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fhi,d^2=hi,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, 114341);