#334530

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

682 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=h,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=dei,e^b=eh,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 241726);