#7696651

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

5045 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7603847);