#7985505

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

5052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=ghi,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, 7892701);