#343388

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
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

739 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:

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

Polycyclic presentation:

2^9|a^2=efh,c^2=i,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,e^b=eh,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250584);