#7709918

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	#59568	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4504 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7617114);