#7786187

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

4830 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)(17,18,21,25)(19,23,27,22)(20,24,29,32)(26,31,28,30);(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,12,16)(17,18,21,25)(19,23,27,22)(20,24,29,32)(26,31,28,30);(2,5)(6,7)(9,11,13,12)(10,15,14,16)(17,19,21,27)(18,22,25,23)(20,28,29,26)(24,31,32,30);(3,8)(6,7)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,29)(30,32);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,16,14,15)(17,20,21,29)(18,24,25,32)(19,26,27,28)(22,30,23,31)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7693383);