#7674161

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	C₂×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4525 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7581357);