#156340

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

2206 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=efi,c^2=eh,d^2=efg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de

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, 63536);