#7741272

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	#38944	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 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)(9,10,15,18)(11,16,12,13)(14,17,22,24)(19,23,20,21)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,13,18,16)(14,20,22,19)(17,23,24,21)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(11,12)(13,16)(19,20)(21,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(9,12,15,11)(10,16,18,13)(14,20,22,19)(17,23,24,21)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,13,15,16)(10,11,18,12)(14,21,22,23)(17,19,24,20)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=i,e^2=g,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, 7648468);