#8388373

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

4877 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=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, 8295569);