#7654496

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	#38909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4772 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,14,12,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31);(1,3,5,6)(2,4,7,8)(9,10)(11,14)(12,15)(13,16)(18,23)(19,25)(26,31)(27,32);(1,3,5,6)(2,4,7,8)(9,11,13,12)(10,14,16,15)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28);(1,3,5,6)(2,4,7,8);(1,2)(3,4)(5,7)(6,8)(9,11,13,12)(10,14,16,15)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,15,16,14)(17,21,22,30)(18,26,23,31)(19,27,25,32)(20,28,24,29)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=gh,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, 7561692);