#156604

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=efg,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 63800);