#333343

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

701 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)(5,6,9,14,10,15,20,13)(7,16,18,8,12,19,17,11)(21,22,24,27)(23,29,31,26)(25,28,32,35)(30,36,33,34);(5,7,10,12)(6,11,15,8)(9,17,20,18)(13,16,14,19)(23,30)(25,32)(26,34)(28,35)(29,36)(31,33);(2,4)(5,8,10,11)(6,12,15,7)(9,19,20,16)(13,18,14,17)(21,23,24,31)(22,26,27,29)(25,33,32,30)(28,36,35,34);(2,4)(6,13)(7,17)(9,20)(12,18)(14,15)(16,19)(22,27)(26,29)(28,35)(34,36)

Polycyclic presentation:

2^9|a^2=ei,b^2=h,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, 240539);