#7739351

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	#34977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5108 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)(3,5)(4,6)(7,8)(9,10)(11,16)(12,14)(13,15);(1,2,5,3)(4,7,8,6)(11,13)(15,16)(18,23)(21,29)(24,25)(27,28);(2,3)(6,7)(9,11,14,13)(10,15,12,16)(17,18)(19,25)(20,24)(21,26)(22,23)(27,32)(28,31)(29,30);(17,19,22,20)(18,24,23,25)(21,27,29,28)(26,31,30,32);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,13,16)(17,20,22,19)(18,25,23,24)(21,28,29,27)(26,32,30,31);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,15,12,16)(17,21,22,29)(18,26,23,30)(19,27,20,28)(24,31,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7646547);