#8095065

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4361 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002261);