#8324114

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	#42591	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4234 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8231310);