#207441

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1004 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)(3,6)(4,5)(7,8)(9,10,14,16,13,11,12,15)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(9,11,14,15,13,10,12,16)(17,19,20,28,21,27,29,26)(18,22,24,32,23,30,31,25);(2,5)(6,8)(10,11)(15,16);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,15,11,16)(18,23)(22,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=ci,e^b=eh,g^b=gh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114637);