#229811

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
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

651 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)(5,6,8,12)(7,14,15,10)(9,11,17,13)(16,18,20,19)(21,22,26,28,25,29,24,33)(23,27,31,36,32,30,34,35);(2,3)(5,7,8,15)(6,10,12,14)(9,16,17,20)(11,19,13,18)(21,22,24,33,25,29,26,28)(23,30,34,36,32,27,31,35);(6,11)(7,15)(9,17)(10,18)(12,13)(14,19)(21,23)(22,27)(24,34)(25,32)(26,31)(28,36)(29,30)(33,35);(1,3,4,2)(6,12)(7,15)(11,13)(16,20)(21,24,25,26)(22,28,29,33)(23,31,32,34)(27,35,30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137007);