#7736674

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	#51165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4476 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)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,31,34)(27,29,28,32)(30,33,39,38)(35,37,36,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,26)(27,29)(28,32)(30,38)(31,34)(33,39)(35,40)(36,37);(11,18)(12,20)(14,15)(16,23)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(1,4,5,3)(2,7,8,6)(9,12,13,20)(10,16,17,23)(11,19,18,24)(14,21,15,22)(25,29,31,32)(26,27,34,28)(30,37,39,40)(33,35,38,36);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,30,31,39)(26,33,34,38)(27,35,28,36)(29,37,32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7643870);