#4038891

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

2174 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)(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,17)(18,20)(21,22,26,23)(24,28,25,27);(6,7)(10,11)(14,16)(15,20)(21,23,26,22)(24,27,25,28);(8,12)(10,11)(14,16)(18,19)(21,24,26,25)(22,27,23,28);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,17,20)(14,18,16,19);(5,8)(6,10)(7,11)(9,12)(13,16,17,14)(15,18,20,19)(21,25,26,24)(22,28,23,27)

Polycyclic presentation:

2^9|a^2=fg,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, 3946087);