#9863081

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
prime radical	#59112	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=gi,e^2=ghi,f^2=h,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, 9770277);