#7791034

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	#47876	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=hi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7698230);