#4038890

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

2062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=f,e^2=g,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, 3946086);