#615574

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2628 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 522770);