#7941718

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

4783 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7848914);