#7954710

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	#42091	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4472 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,e^2=gi,f^2=hi,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, 7861906);