#214601

order	512
nilpotency class	3
isoclinism	isoclinic class of #214000
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1722 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)(3,7,8,5)(4,6)(9,10)(11,21,25,16)(12,18,13,27)(14,20)(15,19)(17,26,35,28)(22,38)(23,32)(24,37,29,33)(30,36)(31,39)(34,40);(9,11)(10,16)(12,22)(13,23)(14,29)(15,25)(17,34)(18,32)(19,21)(20,37)(24,30)(26,40)(27,38)(28,39)(31,35)(33,36);(1,3,4,8)(2,5,6,7)(9,12,15,13)(10,17,19,35)(11,22,25,23)(14,28,30,26)(16,31,21,34)(18,20,27,36)(24,40,29,39)(32,33,38,37);(9,13,15,12)(10,18,19,27)(11,23,25,22)(14,28,30,26)(16,32,21,38)(17,20,35,36)(24,40,29,39)(31,33,34,37)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=ghi,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121797);