#7678842

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

5068 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)(3,6)(4,8)(5,7)(10,13)(14,16)(17,18)(19,24)(20,22)(21,26)(23,25)(27,31)(28,30)(29,32);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,23,20)(18,24,25,22)(21,28,29,27)(26,30,32,31);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(18,25)(19,20)(21,29)(30,31);(1,2)(3,6)(4,8)(5,7)(17,20,23,19)(18,22,25,24)(21,28,29,27)(26,30,32,31);(1,4,5,3)(2,8,7,6)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32);(17,22)(18,19)(20,25)(21,30)(23,24)(26,27)(28,32)(29,31)

Polycyclic presentation:

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

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, 7586038);