#7733365

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

7416 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)(13,14,18,20)(15,16,21,19)(17,22,26,28)(23,24,25,27);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15,18,21)(14,19,20,16)(17,25,26,23)(22,24,28,27);(6,9)(10,11)(14,20)(15,21)(22,28)(23,25);(5,6)(7,10)(8,9)(11,12)(13,16,18,19)(14,21,20,15)(17,24,26,27)(22,25,28,23);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,25)(26,28);(5,7)(6,10)(8,12)(9,11)(13,17)(14,22)(15,23)(16,24)(18,26)(19,27)(20,28)(21,25)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

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, 7640561);