#333301

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×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

792 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)(5,6,11,12,10,7,8,14)(9,17,16,15,19,20,18,13)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,7,10,6)(8,14,11,12)(9,17,19,20)(13,18,15,16)(21,23)(22,26)(24,31)(25,30)(27,29)(28,34)(32,33)(35,36);(2,4)(5,8,10,11)(6,12,7,14)(9,16,19,18)(13,20,15,17)(23,30)(25,33)(26,34)(28,36)(29,35)(31,32);(2,4)(5,9,10,19)(6,13,7,15)(8,16,11,18)(12,20,14,17)(22,27)(23,31)(28,36)(30,32)

Polycyclic presentation:

2^9|a^2=ehi,b^2=h,c^2=h,d^2=h,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, 240497);