#8115876

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	#58411	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8023072);