#206872

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114068);