#7781037

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4371 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)(9,10,15,19)(11,16,14,18)(12,13,17,24)(20,21,22,23)(25,26,30,35)(27,32,37,31)(28,36,39,33)(29,34,40,38);(1,2)(3,7)(4,8)(5,6)(9,11,15,14)(10,16,19,18)(12,20,17,22)(13,21,24,23)(25,27)(26,31)(28,34)(29,36)(30,37)(32,35)(33,40)(38,39);(11,14)(16,18)(20,22)(21,23)(25,27,30,37)(26,31,35,32)(28,34,39,38)(29,33,40,36);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,20)(14,22)(15,17)(16,21)(18,23)(19,24)(25,27)(26,32)(28,38)(29,36)(30,37)(31,35)(33,40)(34,39);(9,13)(10,17)(11,21)(12,19)(14,23)(15,24)(16,22)(18,20)(25,28,30,39)(26,33,35,36)(27,34,37,38)(29,31,40,32);(1,4,5,3)(2,8,6,7)(9,14,15,11)(10,18,19,16)(12,22,17,20)(13,23,24,21)(25,29)(26,34)(27,36)(28,31)(30,40)(32,39)(33,37)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7688233);