#7864670

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4622 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=h,e^2=gh,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, 7771866);