#8031575

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	#39408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4337 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=hi,e^2=i,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, 7938771);