#7711612

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	#39393	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 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)(11,16)(12,13)(14,19)(15,17)(18,24)(20,21)(22,23)(25,26)(27,33)(28,36)(29,35)(30,32)(31,38)(34,37)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,14)(10,16,17,19)(12,20,18,22)(13,21,24,23)(25,27)(26,29)(28,34)(30,35)(31,37)(32,33)(36,39)(38,40);(10,17)(12,18)(16,19)(20,22)(25,26,30,32)(27,33,35,29)(28,36,38,31)(34,37,40,39);(1,3,5,4)(2,7,6,8)(9,12,15,18)(10,13,17,24)(11,20,14,22)(16,21,19,23)(25,26)(27,29)(28,31)(30,32)(33,35)(34,37)(36,38)(39,40);(9,13)(10,18)(11,21)(12,17)(14,23)(15,24)(16,22)(19,20)(25,28,30,38)(26,31,32,36)(27,34,35,40)(29,37,33,39);(1,4,5,3)(2,8,6,7)(9,14,15,11)(10,19,17,16)(12,22,18,20)(13,23,24,21)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,40)(38,39)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=gh,e^2=i,f^2=gh,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, 7618808);