#7813206

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

5084 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)(9,10,13,16)(11,14,12,15)(17,18,23,21)(19,27,28,24)(20,25,29,30)(22,26,32,31);(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,22)(21,27)(23,28)(25,31)(26,30)(29,32);(1,3,5,4)(2,6,7,8)(19,28)(22,32)(24,27)(26,31);(3,4)(6,8)(9,11,13,12)(10,14,16,15)(17,20,23,29)(18,25,21,30)(19,22,28,32)(24,31,27,26);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,15,16,14)(17,21,23,18)(19,27,28,24)(20,25,29,30)(22,31,32,26);(1,2)(3,8)(4,6)(5,7)(17,22)(18,26)(19,29)(20,28)(21,31)(23,32)(24,25)(27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=gh,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, 7720402);