#2039793

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

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fg,d^2=fgh,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, 1946989);