#7658512

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

5141 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565708);