#7714453

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	#50368	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 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)(11,15)(12,20)(13,18)(14,16)(17,22)(19,24)(21,23)(25,26,30,27)(28,31,36,33)(29,34,37,32)(35,39,40,38);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(26,27)(31,33)(32,34)(38,39);(1,2,5,4)(3,6,8,7)(10,16)(15,18)(17,20)(23,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=i,e^2=g,f^2=h,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, 7621649);