#229736

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

675 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,d^c=di,f^c=fh

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, 136932);