#371062

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1124 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)(7,12)(8,10)(9,11)(21,22,25,29,24,28,26,32)(23,27,31,35,30,34,33,36);(5,7,9,8)(6,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,22,26,32,24,28,25,29)(23,27,33,36,30,34,31,35);(2,4)(5,8,9,7)(6,10,11,12)(14,17)(18,20)(21,23,24,30)(22,27,28,34)(25,31,26,33)(29,35,32,36);(6,11)(10,12)(13,15,16,19)(14,18,17,20)(23,30)(27,34)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di

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, 278258);