#7654685

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	#42970	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4647 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=g,e^2=gi,f^2=gi,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, 7561881);