#405617

order	512
nilpotency class	3
isoclinism	isoclinic class of #395246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312813);