#8088009

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	#51966	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4439 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7995205);