#7852858

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4741 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=g,e^2=h,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, 7760054);