#7848812

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	#45064	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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)(9,10,15,12)(11,16,14,18)(13,20,22,17)(19,24,21,23)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,18)(13,19)(14,15)(17,23)(20,24)(21,22)(25,27)(26,31)(28,37)(29,39)(30,36)(32,33)(34,35)(38,40);(1,2,5,4)(3,6,8,7)(10,12)(11,14)(17,20)(19,21)(27,36)(29,37)(31,35)(33,40);(9,12,15,10)(11,18,14,16)(13,17,22,20)(19,23,21,24)(25,28)(26,32)(27,37)(29,36)(30,39)(31,33)(34,38)(35,40);(1,4,5,2)(3,7,8,6)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,26)(27,31)(28,38)(29,33)(30,34)(32,39)(35,36)(37,40);(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,29,30,37)(26,33,34,40)(27,28,36,39)(31,38,35,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7756008);