#650874

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

2274 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)(5,6,10,9)(7,8,12,11)(13,14,19,18)(15,16,17,20)(22,25)(26,27);(6,9)(7,12)(13,15,19,17)(14,20,18,16)(21,22,24,25)(23,27,28,26);(5,7,10,12)(6,11,9,8)(13,16)(14,17)(15,18)(19,20);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,17,19,15)(14,20,18,16);(5,9,10,6)(7,8,12,11)(13,18,19,14)(15,20,17,16)(21,23,24,28)(22,26,25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 558070);