#7768052

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,e^2=g,f^2=h,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, 7675248);