#4047046

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=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, 3954242);