#7657030

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

4651 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,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, 7564226);