#8001026

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4686 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7908222);