#8438511

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

5041 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8345707);