#3598528

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

2712 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,e^2=fgi,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, 3505724);