#2254956

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2673	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fhi,c^2=fgi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2162152);