#7813221

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

4991 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7720417);