#210969

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 118165);