#7987244

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

4197 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)(11,15)(12,20)(13,18)(14,16)(17,22)(19,24)(21,23)(25,26,30,33)(27,28,35,31)(29,32,39,40)(34,36,37,38);(1,2,5,6)(3,7,4,8)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,39)(38,40);(10,16)(15,18)(17,20)(23,24)(25,27,30,35)(26,31,33,28)(29,34,39,37)(32,38,40,36);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,40)(37,39);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,28,30,31)(26,27,33,35)(29,38,39,36)(32,37,40,34);(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,29)(26,32)(27,34)(28,36)(30,39)(31,38)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7894440);