#363437

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1235 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)(3,7)(4,5)(6,8)(9,10,14,20,15,19,13,29)(11,21,24,36,25,16,23,33)(12,26,27,17,28,18,30,35)(22,32,38,34,39,37,40,31);(2,5)(6,7)(9,11,15,25)(10,16,19,21)(12,22,28,39)(13,24,14,23)(17,34,35,31)(18,32,26,37)(20,36,29,33)(27,40,30,38);(9,12)(10,17)(11,22)(13,30)(14,27)(15,28)(16,31)(18,20)(19,35)(21,34)(23,40)(24,38)(25,39)(26,29)(32,33)(36,37);(1,3,4,8)(2,6,5,7)(9,12)(10,18)(11,22)(13,27)(14,30)(15,28)(16,32)(17,20)(19,26)(21,37)(23,38)(24,40)(25,39)(29,35)(31,33)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270633);