#7919930

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=h,e^2=hi,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, 7827126);