#7708417

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

4965 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7615613);