#7918108

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	#45447	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4446 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,e^2=gh,f^2=g,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, 7825304);