#8664997

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

4978 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)(11,15)(12,13)(14,16)(17,18,22,25)(19,24,26,23)(20,29,30,21)(27,28,32,31);(1,2)(3,8)(4,6)(5,7)(10,13)(14,15)(19,26)(23,24)(27,32)(28,31);(1,3,5,6)(2,4,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,25)(29,31)(30,32);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,24)(20,27)(21,31)(22,26)(23,25)(28,29)(30,32);(1,2)(3,4)(5,7)(6,8)(17,20,22,30)(18,21,25,29)(19,27,26,32)(23,28,24,31);(1,4)(2,6)(3,7)(5,8)(17,21)(18,20)(19,28)(22,29)(23,27)(24,32)(25,30)(26,31)

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8572193);