#187470

order	512
nilpotency class	3
isoclinism	isoclinic class of #187410
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62822	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei,f^c=fh,g^c=gh

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, 94666);