#7955028

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	#37701	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=g,e^2=gi,f^2=i,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, 7862224);