#7968408

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

5517 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)(3,8)(4,7)(5,6)(9,10,13,16)(11,12,14,15)(17,18,23,27)(19,24,29,31)(20,21,30,25)(22,26,28,32);(2,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,19,23,29)(18,24,27,31)(20,22,30,28)(21,26,25,32);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,16)(13,15)(17,20,23,30)(18,25,27,21)(19,28,29,22)(24,26,31,32);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,16)(13,14)(17,20)(18,25)(19,22)(21,27)(23,30)(24,32)(26,31)(28,29);(1,4,5,3)(2,7,6,8)(9,11)(10,12)(13,14)(15,16)(17,21,23,25)(18,20,27,30)(19,26,29,32)(22,31,28,24);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,16)(13,15)(17,22)(18,26)(19,20)(21,24)(23,28)(25,31)(27,32)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7875604);