#7812344

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

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

5524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,4)(5,6)(7,8)(9,10,13,14)(11,15,16,12)(17,18)(19,23)(20,25)(21,30)(22,24)(26,31)(27,28)(29,32);(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,21)(22,27)(24,28)(25,30)(26,32)(29,31);(2,6)(3,7)(10,14)(12,15)(18,24)(19,27)(20,29)(26,30);(1,3,5,7)(2,4,6,8)(9,12)(10,11)(13,15)(14,16)(17,20,22,29)(18,25,24,32)(19,21,27,31)(23,30,28,26);(1,4)(2,7)(3,6)(5,8)(9,11)(10,12)(13,16)(14,15)(17,21)(18,26)(19,20)(22,31)(23,32)(24,30)(25,28)(27,29);(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31)

Polycyclic presentation:

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

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, 7719540);