#7658133

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	#37553	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4238 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7565329);