#613028

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=g,e^2=f,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, 520224);