#160321

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=fg,c^2=efgh,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,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, 67517);