#928615

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2044 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=fhi,e^2=g,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, 835811);