#7715448

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	#38035	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4193 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)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(26,30)(28,36)(31,33)(34,40);(1,3,5,8)(2,6,4,7)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(11,12)(15,18)(19,20)(23,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,4,5,2)(3,7,8,6)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(9,13,14,22)(10,16,17,21)(11,19,12,20)(15,23,18,24)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622644);