#8765600

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5099 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8672796);