#7696473

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

4848 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)(4,8)(5,6)(10,14)(15,16)(17,18)(19,21)(20,25)(22,24)(23,26)(27,30)(28,32)(29,31);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,27)(21,24)(22,26)(25,28)(29,32)(30,31);(2,6)(7,8)(9,11,13,12)(10,15,14,16)(18,24)(19,26)(25,31)(27,32);(1,2)(3,7)(4,8)(5,6)(17,19)(18,23)(20,27)(21,24)(22,26)(25,28)(29,32)(30,31);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,16,14,15)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32);(1,4,5,3)(2,8,6,7)(17,21,22,23)(18,19,24,26)(20,28,29,30)(25,32,31,27)

Polycyclic presentation:

2^9|c^2=gi,e^2=gh,f^2=h,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, 7603669);