#7656555

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	#42881	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4627 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)(10,14)(15,16)(17,18)(19,21)(20,25)(22,27)(23,28)(24,26)(29,31)(30,32);(1,3,6,4)(2,7,5,8)(9,10,13,14)(11,15,12,16)(17,19,23,26)(18,24,28,21)(20,30,31,27)(22,29,32,25);(1,3,6,4)(2,7,5,8)(9,11,13,12)(10,15,14,16)(17,20,23,31)(18,25,28,29)(19,27,26,30)(21,22,24,32);(9,11,13,12)(10,15,14,16)(17,21,23,24)(18,26,28,19)(20,22,31,32)(25,30,29,27);(1,4,6,3)(2,7,5,8)(9,11,13,12)(10,15,14,16)(17,22)(18,27)(19,29)(20,24)(21,31)(23,32)(25,26)(28,30);(1,5,6,2)(3,8,4,7)(9,12,13,11)(10,16,14,15)(17,18)(19,24)(20,25)(21,26)(22,30)(23,28)(27,32)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gh,e^2=gi,f^2=gi,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, 7563751);