#8923653

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	#59053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4351 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,7)(3,4,8,6)(25,26,30,32)(27,31,35,39)(28,33,34,29)(36,37,38,40);(1,3,5,8)(2,6,7,4)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27)(26,31)(28,38)(29,37)(30,35)(32,39)(33,40)(34,36);(1,2)(3,4)(5,7)(6,8)(10,11)(13,21)(15,17)(19,24)(26,32)(27,35)(29,33)(36,38);(1,4)(2,3)(5,6)(7,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27)(26,31)(28,38)(29,37)(30,35)(32,39)(33,40)(34,36);(1,3)(2,4)(5,8)(6,7)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,34)(26,33,32,29)(27,36,35,38)(31,37,39,40);(1,4)(2,3)(5,6)(7,8)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,33)(26,34,32,28)(27,37,35,40)(31,36,39,38)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8830849);