#7838654

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	#44762	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4233 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)(11,12)(13,17)(14,15)(16,20)(18,19)(21,24)(22,23)(25,26,31,35)(27,32,30,34)(28,29,33,40)(36,37,38,39);(1,3,5,8)(2,6,4,7)(9,11,14,16)(10,12,15,20)(13,21,22,18)(17,24,23,19)(25,27)(26,32)(28,36)(29,37)(30,31)(33,38)(34,35)(39,40);(10,15)(11,16)(13,22)(19,24)(27,30)(32,34)(36,38)(37,39);(1,2,5,4)(3,6,8,7)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,24)(22,23)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(1,2,5,4)(3,6,8,7)(9,12)(10,16)(11,15)(13,19)(14,20)(17,21)(18,23)(22,24)(25,29)(26,33)(27,37)(28,35)(30,39)(31,40)(32,38)(34,36);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,17,15,23)(11,18,16,21)(12,19,20,24)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,39,40,37)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,e^2=g,f^2=gi,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, 7745850);