#4344121

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
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

1860 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)(6,7)(10,11)(14,17)(15,19)(21,22,27,32)(23,29,24,28)(25,34,35,30)(26,31,36,33);(14,17)(18,20)(21,23,27,24)(22,28,32,29)(25,33,35,31)(26,34,36,30);(5,6,9,7)(8,10,12,11)(13,14,16,17)(15,18,19,20)(21,24,27,23)(22,29,32,28)(25,33,35,31)(26,34,36,30);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,19)(17,20)(21,25)(22,30)(23,33)(24,31)(26,28)(27,35)(29,36)(32,34);(5,8,9,12)(6,10,7,11)(13,15)(14,18)(16,19)(17,20)(21,26,27,36)(22,31,32,33)(23,30,24,34)(25,28,35,29)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=fhi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4251317);