#7646894

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	#42658	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4545 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)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,30)(21,28)(22,26)(23,31)(24,27)(29,32);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,12,16)(17,19,24,29)(18,25,27,32)(20,28,22,23)(21,26,31,30);(9,12,14,10)(11,15,13,16)(17,20,24,22)(18,26,27,30)(19,28,29,23)(21,25,31,32);(9,13,14,11)(10,15,12,16)(17,21)(18,23)(19,26)(20,25)(22,32)(24,31)(27,28)(29,30);(17,22,24,20)(18,26,27,30)(19,23,29,28)(21,25,31,32);(1,3,4,7)(2,6,5,8)(17,23)(18,21)(19,20)(22,29)(24,28)(25,26)(27,31)(30,32)

Polycyclic presentation:

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

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, 7554090);