#7696358

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

4764 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=g,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, 7603554);