#7985551

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

5000 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,e^2=g,f^2=i,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, 7892747);