#8215372

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	#55593	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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8122568);