#288128

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 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)(7,15)(8,11)(9,19)(10,12)(13,17)(14,16)(18,20)(21,22,25,28)(23,26,30,34)(24,27,31,35)(29,33,32,36);(1,3,4,2)(5,7,16,18,10,17,9,8)(6,11,19,13,12,20,14,15)(23,29)(24,31)(26,33)(27,35)(30,32)(34,36);(2,3)(6,12)(7,17)(8,18)(14,19)(21,23,25,30)(22,26,28,34)(24,32,31,29)(27,36,35,33);(5,8,9,17,10,18,16,7)(6,13,14,11,12,15,19,20)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ef,c^2=i,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,d^c=dh,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, 195324);