#7947441

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	#35393	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4415 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)(9,10,14,16)(11,15,19,21)(12,20,22,17)(13,18,23,24)(25,26,30,34)(27,29,33,31)(28,36,38,32)(35,39,40,37);(1,3,5,8)(2,6,4,7)(9,11,14,19)(10,15,16,21)(12,18,22,24)(13,20,23,17)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,37)(38,40);(10,16)(11,19)(17,20)(18,24)(25,26,30,34)(27,29,33,31)(28,32,38,36)(35,37,40,39);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,22)(23,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(9,12)(10,17)(11,18)(13,15)(14,22)(16,20)(19,24)(21,23)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,39)(38,40);(9,13)(10,18)(11,20)(12,21)(14,23)(15,22)(16,24)(17,19)(25,29)(26,33)(27,34)(28,37)(30,31)(32,40)(35,36)(38,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7854637);