#384064

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×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

1092 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)(7,9)(8,11)(10,12)(13,14)(15,17)(16,18)(19,20)(21,22,26,29,25,27,24,32)(23,28,30,36,31,34,33,35);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15,16,19)(14,17,18,20)(22,27)(28,34)(29,32)(35,36);(2,4)(5,8,10,9)(6,11,12,7)(14,18)(17,20)(21,23,25,31)(22,28,27,34)(24,33,26,30)(29,36,32,35);(5,9,10,8)(6,11,12,7)(14,18)(17,20)(21,24,25,26)(22,29,27,32)(23,30,31,33)(28,35,34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=dei,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, 291260);