#7718427

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4366 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)(12,14)(13,20)(16,21)(17,18)(19,24)(22,23)(25,26,30,27)(28,31,35,33)(29,34,37,32)(36,39,40,38);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24)(26,27)(28,35)(32,34)(36,40);(1,4,5,2)(3,7,8,6)(10,12)(11,17)(16,20)(19,23)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39);(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24);(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7625623);