#4101921

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

1740 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)(7,12)(9,11)(14,17)(15,19)(21,22,26,23)(24,29,31,27)(25,28,34,30)(32,36,33,35);(5,6,10,8)(7,9,12,11)(15,19)(18,20)(22,23)(24,31)(28,30)(32,33);(5,7,10,12)(6,11,8,9)(13,14)(15,18)(16,17)(19,20)(21,23,26,22)(24,29,31,27)(25,28,34,30)(32,35,33,36);(1,2,3,4)(5,8,10,6)(7,9,12,11)(21,24,26,31)(22,27,23,29)(25,33,34,32)(28,36,30,35);(5,9)(6,7)(8,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,25,26,34)(22,28,23,30)(24,32,31,33)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,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=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, 4009117);