#7864198

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
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4673 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7771394);