#7736398

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

5401 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,16)(14,15)(17,18,23,20)(19,25,21,24)(22,26,30,28)(27,32,29,31);(1,2,5,3)(4,7,8,6)(9,11,14,12)(10,13,15,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32);(2,3)(6,7)(10,15)(11,12)(18,20)(24,25)(26,28)(31,32);(17,20,23,18)(19,25,21,24)(22,28,30,26)(27,32,29,31);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,13,15,16)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31);(1,4,5,8)(2,6,3,7)(9,13)(10,11)(12,15)(14,16)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=hi,e^2=h,f^2=g,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, 7643594);