#7642170

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

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

8606 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549366);