#7722691

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

4838 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,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, 7629887);