#8039015

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

5312 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)(3,4)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,22,23)(19,25,28,24)(20,26,29,32)(21,30,31,27);(1,3,6,7)(2,4,5,8)(9,11,13,16)(10,14,12,15)(18,23)(20,29)(24,25)(27,30);(1,2,6,5)(3,8,7,4)(17,19,22,28)(18,24,23,25)(20,27,29,30)(21,26,31,32);(1,4)(2,7)(3,5)(6,8)(17,19)(18,25)(20,27)(21,26)(22,28)(23,24)(29,30)(31,32);(1,4)(2,3)(5,7)(6,8)(17,20)(18,26)(19,27)(21,25)(22,29)(23,32)(24,31)(28,30);(1,5,6,2)(3,8,7,4)(9,12,13,10)(11,15,16,14)(17,21)(18,27)(19,26)(20,25)(22,31)(23,30)(24,29)(28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7946211);