#7919666

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

4124 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=gh,e^2=g,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, 7826862);