#156590

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=e,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, 63786);