#7786310

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	#49545	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5038 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=gh,f^2=g,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, 7693506);