#4433706

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3399 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)(6,7)(10,11)(13,14)(15,18)(16,19)(17,20)(22,26)(24,27);(5,6,9,7)(8,10,12,11)(13,15)(14,17)(16,20)(18,19)(21,22,25,26)(23,24,28,27);(13,15)(14,17)(16,20)(18,19)(21,23)(22,27)(24,26)(25,28);(1,2,3,4)(5,7,9,6)(8,10,12,11)(13,15)(14,18)(16,20)(17,19)(21,23)(22,24)(25,28)(26,27);(5,8)(6,10)(7,11)(9,12)(13,14)(15,17)(16,19)(18,20)(21,24)(22,28)(23,26)(25,27)

Polycyclic presentation:

2^9|b^2=fg,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,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, 4340902);