#786590

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2589 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=fh,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 693786);