#8063105

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

4742 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)(3,4)(5,6)(7,8)(9,10,13,16)(11,14,12,15)(17,18,23,25)(19,28,22,24)(20,21,27,26)(29,31,32,30);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19,23,22)(18,24,25,28)(20,29,27,32)(21,30,26,31);(3,7)(4,8)(9,11,13,12)(10,14,16,15)(18,25)(19,22)(21,26)(29,32);(2,6)(4,8)(9,11,13,12)(10,14,16,15)(17,20,23,27)(18,26,25,21)(19,29,22,32)(24,31,28,30);(1,4)(2,3)(5,8)(6,7)(9,11,13,12)(10,14,16,15)(17,21,23,26)(18,27,25,20)(19,30,22,31)(24,32,28,29);(1,2)(3,4)(5,6)(7,8)(9,12,13,11)(10,15,16,14)(17,22,23,19)(18,24,25,28)(20,29,27,32)(21,31,26,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7970301);