#206985

order	512
nilpotency class	3
isoclinism	isoclinic class of #161742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 114181);