#8108328

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

5312 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015524);