#2107610

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1962 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)(7,12)(8,11)(9,10)(14,15)(16,20)(21,22,26,24)(23,27,31,29)(25,28,34,32)(30,35,33,36);(5,7)(6,8)(9,12)(10,11)(13,14,17,15)(16,18,20,19)(22,24)(27,29)(28,32)(35,36);(6,9)(7,11)(13,15,17,14)(16,18,20,19)(21,23,26,31)(22,27,24,29)(25,33,34,30)(28,36,32,35);(1,2)(3,4)(5,8,10,12)(6,11,9,7)(21,24,26,22)(23,29,31,27)(25,32,34,28)(30,36,33,35);(5,9,10,6)(7,12,11,8)(13,16)(14,18)(15,19)(17,20)(21,25,26,34)(22,28,24,32)(23,30,31,33)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=fh,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2014806);