#9061453

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

4913 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)(11,12)(13,15)(14,16)(17,18,21,22)(19,23,25,26)(20,28,29,24)(27,32,31,30);(1,2,5,6)(3,7,8,4)(9,10,14,16)(11,12,13,15)(18,22)(23,26)(24,28)(30,32);(1,2)(3,7)(4,8)(5,6)(11,13)(12,15)(18,22)(19,25)(20,29)(30,32);(2,6)(3,8)(9,11,14,13)(10,15,16,12)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,3)(2,4)(5,8)(6,7)(9,12,14,15)(10,11,16,13)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31);(1,4)(2,3)(5,7)(6,8)(9,13,14,11)(10,15,16,12)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8968649);