#7696518

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

4822 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)(9,10,13,12)(11,15,16,14)(17,18,22,24)(19,21,26,23)(20,25,29,31)(27,30,32,28);(9,11,13,16)(10,14,12,15)(17,19,22,26)(18,23,24,21)(20,27,29,32)(25,28,31,30);(2,6)(7,8)(9,10,13,12)(11,14,16,15)(18,24)(19,26)(25,31)(27,32);(1,2)(3,7)(4,8)(5,6)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32);(1,3,5,4)(2,7,6,8)(9,12,13,10)(11,15,16,14)(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)(18,26,24,19)(20,28,29,30)(25,27,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 7603714);