#2039286

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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

1929 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)(5,6,9,10)(7,11,8,12)(13,14)(15,16)(17,18)(19,20)(22,26)(23,29)(28,32)(31,35);(6,10)(11,12)(14,17)(15,19)(21,22,25,26)(23,30,29,27)(24,32,33,28)(31,34,35,36);(5,7,9,8)(6,11,10,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,27)(24,28)(25,26)(29,30)(31,34)(32,33)(35,36);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(13,16,18,20)(14,19,17,15)(21,23,25,29)(22,27,26,30)(24,31,33,35)(28,34,32,36);(5,6)(7,11)(8,12)(9,10)(13,17,18,14)(15,20,19,16)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=fh,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

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, 1946482);