#8505939

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	#45057	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4457 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413135);