#333299

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×#1621	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

847 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)(5,6,10,11,9,14,7,12)(8,16,15,13,18,19,17,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(6,11)(7,10)(8,17)(12,14)(13,20)(15,18)(21,23)(22,26)(24,31)(25,30)(27,29)(28,34)(32,33)(35,36);(2,4)(5,7,9,10)(6,12,14,11)(8,15,18,17)(13,19,20,16)(23,30)(25,33)(26,34)(28,36)(29,35)(31,32);(2,4)(5,8,9,18)(6,13,14,20)(7,15,10,17)(11,16,12,19)(22,27)(23,31)(28,36)(30,32)

Polycyclic presentation:

2^9|a^2=ehi,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, 240495);