#7632551

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5227 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539747);