#384447

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, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

954 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,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, 291643);