#7714967

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4076 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gi,e^2=ghi,f^2=gh,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, 7622163);