#7966967

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	#59093	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4746 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)(3,5)(4,7)(6,8)(9,10,13,16)(11,14,12,15)(17,18,23,27)(19,21,29,24)(20,30,32,25)(22,26,28,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19,23,29)(18,24,27,21)(20,22,32,28)(25,26,30,31);(1,2)(3,6)(4,7)(5,8)(17,20,23,32)(18,25,27,30)(19,28,29,22)(21,26,24,31);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,22)(21,31)(23,32)(24,26)(27,30)(28,29);(1,3)(2,5)(4,8)(6,7)(9,12,13,11)(10,15,16,14)(17,21)(18,19)(20,31)(22,25)(23,24)(26,32)(27,29)(28,30);(1,2)(3,6)(4,7)(5,8)(17,22)(18,26)(19,20)(21,30)(23,28)(24,25)(27,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7874163);