#411136

order	512
nilpotency class	3
isoclinism	isoclinic class of #395151
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

994 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=gh,c^2=fgi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318332);