#8701553

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

4933 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608749);