#229756

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136952);