#8799112

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

4785 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gi,e^2=i,f^2=h,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, 8706308);