#7763426

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	#37616	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

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, 7670622);