#7670702

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
prime radical	#59009	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4987 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577898);