#9794221

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

4746 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9701417);