#7658452

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	#35771	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4517 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)(9,10,12,14)(11,13,15,16)(17,18,23,24)(19,25,29,32)(20,22,28,26)(21,27,30,31);(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(18,24)(19,29)(20,28)(27,31);(1,2,4,5)(3,6,7,8)(17,19)(18,25)(20,30)(21,28)(22,31)(23,29)(24,32)(26,27);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,20,23,28)(18,26,24,22)(19,30,29,21)(25,27,32,31);(17,21,23,30)(18,27,24,31)(19,28,29,20)(22,25,26,32);(17,22,23,26)(18,28,24,20)(19,31,29,27)(21,32,30,25)

Polycyclic presentation:

2^9|a^2=g,c^2=h,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, 7565648);