#7744645

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	#37269	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4532 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)(9,10,14,13)(11,15,18,16)(12,17,20,21)(19,22,24,23)(25,26)(27,33)(28,30)(29,32)(31,35)(34,39)(36,37)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,16)(14,18)(17,22)(20,24)(21,23)(25,26,30,28)(27,31,35,33)(29,36,37,32)(34,39,40,38);(1,3,5,4)(2,6,8,7)(11,18)(12,20)(15,16)(17,21)(26,28)(31,33)(32,36)(38,39);(1,4,5,3)(2,7,8,6)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,24)(21,22)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,12,14,20)(10,17,13,21)(11,19,18,24)(15,22,16,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(9,13,14,10)(11,15,18,16)(12,21,20,17)(19,22,24,23)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7651841);