#7689347

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

5448 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,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, 7596543);