#288352

order	512
nilpotency class	3
isoclinism	isoclinic class of #264363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ 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)(8,18)(9,19)(10,12)(11,16)(14,15)(17,20)(21,22,24,26)(23,29,31,27)(25,28,32,34)(30,35,33,36);(1,3,4,2)(5,7,15,8,10,17,9,16)(6,11,19,20,12,18,14,13)(22,26)(23,30)(25,32)(27,35)(29,36)(31,33);(2,3)(6,12)(11,18)(13,20)(14,19)(21,23,24,31)(22,27,26,29)(25,33,32,30)(28,35,34,36);(5,8,15,17,10,16,9,7)(6,13,19,11,12,20,14,18)(22,26)(27,29)(28,34)(35,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,c^2=i,d^2=ehi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=gi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195548);