#156363

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

2130 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)(14,17)(15,19)(21,22)(23,26)(24,25)(27,28)(29,30,33,32)(31,34,35,36);(1,3,4,2)(22,25)(26,27)(31,35)(34,36);(5,6,8,10)(7,9,11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,25)(26,28)(29,31)(30,34)(32,36)(33,35);(5,7)(6,9)(8,11)(10,12)(13,15,16,19)(14,18,17,20)(21,23)(22,26)(24,28)(25,27)(29,32,33,30)(31,36,35,34)

Polycyclic presentation:

2^9|a^2=i,b^2=egi,c^2=fh,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, 63559);