#333341

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

782 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)(3,4)(5,6,8,12,9,13,16,11)(7,10,14,18,15,19,20,17)(21,22,24,27)(23,29,31,26)(25,28,32,35)(30,36,33,34);(5,6)(7,10)(8,11)(9,13)(12,16)(14,17)(15,19)(18,20)(23,30)(25,32)(26,34)(28,35)(29,36)(31,33);(2,4)(5,7,9,15)(6,10,13,19)(8,14,16,20)(11,17,12,18)(21,23,24,31)(22,26,27,29)(25,33,32,30)(28,36,35,34);(2,4)(6,11)(8,16)(10,17)(12,13)(14,20)(18,19)(22,27)(26,29)(28,35)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240537);