#387333

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1030 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)(6,9)(7,11)(13,14)(15,19)(16,18)(17,20)(21,22)(23,31)(24,33)(25,30)(26,27)(28,34)(29,32)(35,36);(5,6,8,9)(7,12,11,10)(13,15)(14,17)(16,20)(18,19)(21,23,25,26)(22,27,30,31)(24,34,35,32)(28,33,29,36);(14,18)(15,20)(21,24,23,32,25,35,26,34)(22,28,27,36,30,29,31,33);(5,7,8,11)(6,10,9,12)(15,20)(17,19)(21,24,23,32,25,35,26,34)(22,29,27,33,30,28,31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=fg,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294529);