#334522

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	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

704 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=h,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, 241718);