#7919992

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

4231 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=i,e^2=h,f^2=i,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, 7827188);