#7710978

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	#39836	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4364 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)(10,15)(13,23)(16,22)(17,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(10,15)(12,21)(16,22)(18,19)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(1,3,5,4)(2,7,6,8)(9,11,14,20)(10,16,15,22)(12,19,21,18)(13,17,23,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,19)(13,22)(14,21)(15,24)(16,23)(18,20)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,18,15,19)(11,17,20,24)(12,22,21,16)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618174);