#7657736

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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4540 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564932);