#7655852

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	#35003	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4716 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)(3,8)(4,6)(5,7)(9,10,13,12)(11,15,16,14)(17,18,23,25)(19,26,22,24)(20,21,30,32)(27,28,29,31);(9,11,13,16)(10,14,12,15)(17,19,23,22)(18,24,25,26)(20,29,30,27)(21,28,32,31);(1,3,5,6)(2,4,7,8)(17,20)(18,21)(19,27)(22,29)(23,30)(24,31)(25,32)(26,28);(17,21)(18,20)(19,28)(22,31)(23,32)(24,29)(25,30)(26,27);(1,2)(3,4)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,20)(18,21)(19,29)(22,27)(23,30)(24,28)(25,32)(26,31);(1,4)(2,6)(3,7)(5,8)(17,22,23,19)(18,24,25,26)(20,29,30,27)(21,31,32,28)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7563048);