#7876329

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	#36078	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5522 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=ghi,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, 7783525);