#7932212

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4122 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)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26,31,35)(27,34,29,32)(28,37,30,33)(36,39,38,40);(1,2)(3,6)(4,7)(5,8)(25,27,31,29)(26,32,35,34)(28,38,30,36)(33,40,37,39);(1,3,5,4)(2,6,8,7)(10,15)(11,16)(17,22)(18,23)(25,28,31,30)(26,33,35,37)(27,36,29,38)(32,39,34,40);(1,3,5,4)(2,6,8,7)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,29,31,27)(26,34,35,32)(28,36,30,38)(33,39,37,40);(1,3,5,4)(2,6,8,7)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,26)(27,32)(28,33)(29,34)(30,37)(31,35)(36,39)(38,40);(1,4,5,3)(2,7,8,6)(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,30,31,28)(26,33,35,37)(27,36,29,38)(32,40,34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=ghi,e^2=gi,f^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, 7839408);