#7983258

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

4696 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)(10,14)(15,16)(17,18,21,24)(19,22,25,29)(20,23,28,32)(26,30,27,31);(1,3)(2,4)(5,7)(6,8)(9,10,13,14)(11,15,12,16)(17,19)(18,22)(20,27)(21,25)(23,31)(24,29)(26,28)(30,32);(2,6)(3,7)(19,25)(22,29)(26,27)(30,31);(1,2,5,6)(3,4,7,8)(9,11,13,12)(10,15,14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(1,3)(2,4)(5,7)(6,8)(17,19)(18,22)(20,27)(21,25)(23,31)(24,29)(26,28)(30,32);(1,4)(2,7)(3,6)(5,8)(9,12,13,11)(10,16,14,15)(17,20,21,28)(18,23,24,32)(19,26,25,27)(22,30,29,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,f^2=gi,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, 7890454);