#7687717

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

5208 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)(3,7)(4,5)(6,8)(9,10,13,15)(11,12,16,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,30)(31,32);(1,3,6,4)(2,7,8,5)(9,11)(10,14)(12,15)(13,16)(17,19,22,29)(18,23,26,27)(20,25,30,32)(21,28,31,24);(1,4,6,3)(2,7,8,5)(9,12,13,14)(10,11,15,16)(17,18,22,26)(19,27,29,23)(20,24,30,28)(21,25,31,32);(1,5,6,7)(2,4,8,3)(9,11)(10,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,30)(31,32);(17,20)(18,24)(19,25)(21,23)(22,30)(26,28)(27,31)(29,32);(17,21)(18,25)(19,28)(20,27)(22,31)(23,30)(24,29)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7594913);