#7654042

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

4846 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)(3,8)(4,6)(5,7)(10,14)(15,16)(17,18,24,22)(19,27,23,25)(20,21,26,31)(28,29,30,32);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,15,12,16)(17,19,24,23)(18,25,22,27)(20,30,26,28)(21,29,31,32);(1,3,5,4)(2,6,7,8)(17,20)(18,21)(19,28)(22,31)(23,30)(24,26)(25,32)(27,29);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,21,24,31)(18,26,22,20)(19,29,23,32)(25,28,27,30);(1,2)(3,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16)(17,22,24,18)(19,27,23,25)(20,31,26,21)(28,29,30,32);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,16,14,15)(17,23,24,19)(18,25,22,27)(20,30,26,28)(21,32,31,29)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561238);