#288060

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
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

660 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)(5,6,10,14)(7,12,16,20)(8,17,15,11)(9,19,18,13)(21,22,25,28)(23,26,30,34)(24,27,31,35)(29,33,32,36);(1,3,4,2)(5,7,9,15,10,16,18,8)(6,11,13,12,14,17,19,20)(23,29)(24,31)(26,33)(27,35)(30,32)(34,36);(2,3)(21,23,25,30)(22,26,28,34)(24,32,31,29)(27,36,35,33);(5,8,18,16,10,15,9,7)(6,12,19,11,14,20,13,17)

Polycyclic presentation:

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

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, 195256);