#7784311

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

4974 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)(19,28)(20,25)(21,29)(22,27)(23,30)(24,31)(26,32);(1,2,5,4)(3,6,8,7)(9,10,13,14)(11,15,12,16)(17,19,22,30)(18,23,27,28)(20,26,31,29)(21,24,32,25);(2,4)(6,7)(17,20,22,31)(18,24,27,25)(19,29,30,26)(21,28,32,23);(1,3,5,8)(2,6,4,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,26)(21,23)(22,31)(24,27)(28,32)(29,30);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,24)(20,28)(22,32)(23,31)(25,30)(27,29);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,16,14,15)(17,18)(19,23)(20,25)(21,26)(22,27)(24,31)(28,30)(29,32)

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7691507);