#7846821

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	#52278	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4252 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)(9,10,15,13)(11,16,14,18)(12,17,21,22)(19,23,20,24)(25,26,31,33)(27,32,37,40)(28,34,35,29)(30,36,39,38);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,19)(13,18)(14,15)(17,23)(20,21)(22,24)(25,27,31,37)(26,32,33,40)(28,38,35,36)(29,39,34,30);(1,3,5,4)(2,7,6,8)(11,14)(16,18)(19,20)(23,24)(26,33)(27,37)(28,35)(30,39);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,19)(13,22)(14,20)(15,21)(16,23)(18,24)(25,28)(26,34)(27,38)(29,33)(30,32)(31,35)(36,37)(39,40);(9,13,15,10)(11,18,14,16)(12,17,21,22)(19,23,20,24)(25,29,31,34)(26,35,33,28)(27,39,37,30)(32,36,40,38);(9,14,15,11)(10,18,13,16)(12,20,21,19)(17,24,22,23)(25,30,31,39)(26,36,33,38)(27,34,37,29)(28,32,35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=g,e^2=i,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, 7754017);