#7664913

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	#38908	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4699 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)(3,4)(5,7)(6,8)(9,10,13,16)(11,14,12,15)(19,25)(20,27)(22,29)(23,28);(1,3,5,6)(2,4,7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(1,4)(2,6)(3,7)(5,8)(9,11,13,12)(10,14,16,15)(17,18,21,24)(19,22,25,29)(20,23,27,28)(26,30,31,32);(9,11,13,12)(10,14,16,15)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,31)(28,32);(1,2)(3,4)(5,7)(6,8)(9,11,13,12)(10,14,16,15)(17,18)(19,22)(20,28)(21,24)(23,27)(25,29)(26,32)(30,31);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,15,16,14)(17,20)(18,23)(19,26)(21,27)(22,30)(24,28)(25,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572109);