#7645974

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	#39048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4556 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)(13,21)(17,24)(19,23)(20,22)(25,26)(27,31)(28,36)(29,35)(30,33)(32,38)(34,37)(39,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,27,30,35)(26,31,33,29)(28,34,38,40)(32,39,36,37);(1,3,5,4)(2,6,8,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40);(9,11)(10,15)(12,18)(13,19)(14,16)(17,22)(20,24)(21,23)(25,29)(26,27)(28,37)(30,31)(32,34)(33,35)(36,40)(38,39);(9,12,14,15)(10,16,18,11)(13,20,21,22)(17,23,24,19)(25,26)(27,31)(28,32)(29,35)(30,33)(34,39)(36,38)(37,40);(1,4,5,3)(2,7,8,6)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7553170);