#3345404

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fgh,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3252600);