#7664686

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
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7390 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)(15,21)(16,23)(19,25)(20,24);(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,20)(17,18)(21,25)(22,26)(23,24)(27,28);(1,2)(3,4)(14,18)(19,25)(20,24)(26,28);(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,27)(24,28);(1,2,3,4)(13,14)(15,19)(16,24)(17,18)(20,23)(21,25)(22,28)(26,27);(1,2)(3,4)(13,16)(14,20)(15,22)(17,23)(18,24)(19,26)(21,27)(25,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7571882);