#364357

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1350 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:

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

Polycyclic presentation:

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

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, 271553);