#158256

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
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

2300 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=fgh,d^2=efg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 65452);