#8919408

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	#47869	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4186 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,e^2=ghi,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, 8826604);