#7652370

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7559566);