#7653874

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	#37439	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=g,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, 7561070);