#7644346

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

9322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 7551542);