#7708416

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

4969 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7615612);