#947348

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 854544);