#160087

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

2864 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)(10,12)(14,17)(15,19)(21,22,24,26)(23,25,27,28)(29,30,32,33)(31,34,35,36);(2,4)(6,9)(7,11)(14,17)(15,19)(23,27)(25,28)(30,33)(31,35);(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28)(29,30)(31,36)(32,33)(34,35);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,23,24,27)(22,25,26,28)(29,31)(30,34)(32,35)(33,36)

Polycyclic presentation:

2^9|a^2=efg,c^2=egh,d^2=ef,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, 67283);