#8125469

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	#45255	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4307 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=i,e^2=h,f^2=gi,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, 8032665);