#7898057

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	C₂×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4353 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7805253);