#160114

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2216 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fgh,d^2=f,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, 67310);