#8095942

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	#52294	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4502 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:

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

Polycyclic presentation:

2^9|a^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003138);