#1614426

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2182 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)(5,6,8,10)(7,11,12,9)(13,14,17,20)(15,16,19,18)(22,26)(23,29)(24,32)(34,36);(7,12)(9,11)(15,19)(16,18)(21,22,25,26)(23,27,29,30)(24,33,32,28)(31,36,35,34);(5,6)(7,9)(8,10)(11,12)(13,15,17,19)(14,18,20,16)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(1,2,3,4)(5,7,8,12)(6,9,10,11)(13,16,17,18)(14,19,20,15)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=h,d^2=fhi,e^2=f,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, 1521622);