#10368476

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	#59005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4597 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:

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

Polycyclic presentation:

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

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, 10275672);