#1634931

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2150 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fghi,d^2=fg,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1542127);