#8110530

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

4996 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,15)(14,16)(18,23)(20,29)(24,30)(27,32);(2,6)(4,7)(10,11)(14,15)(19,26)(24,30)(27,32)(28,31);(1,3)(2,4)(5,8)(6,7)(9,10,13,11)(12,14,16,15)(17,18,22,23)(19,24,26,30)(20,25,29,21)(27,31,32,28);(1,4)(2,3)(5,7)(6,8)(9,11,13,10)(12,15,16,14)(17,19,22,26)(18,24,23,30)(20,27,29,32)(21,28,25,31);(1,4)(2,3)(5,7)(6,8)(9,12,13,16)(10,14,11,15)(17,20)(18,25)(19,27)(21,23)(22,29)(24,31)(26,32)(28,30);(1,3)(2,4)(5,8)(6,7)(17,21,22,25)(18,20,23,29)(19,28,26,31)(24,27,30,32)

Polycyclic presentation:

2^9|c^2=ghi,d^2=ghi,e^2=i,f^2=gh,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, 8017726);