#7850311

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	#59977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4709 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7757507);