#7722499

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

4735 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7629695);