#182905

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,9,6,10,4,7)(3,8,13,15,11,16,12,14)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,42)(30,43)(32,38)(33,39)(34,41)(35,40)(37,45)(44,48)(46,47);(2,7,10,9)(3,11)(8,15,16,14)(12,13)(17,19,22,32,23,31,21,30)(18,24,26,38,27,28,36,43)(20,29,33,44,34,45,35,46)(25,37,40,48,41,42,39,47);(1,3,5,13,6,11,4,12)(2,8,9,15,10,16,7,14)(17,20,23,34)(18,25,27,41)(19,29,31,45)(21,35,22,33)(24,37,28,42)(26,40,36,39)(30,46,32,44)(38,48,43,47);(17,21,23,22)(18,26,27,36)(19,30,31,32)(20,33,34,35)(24,38,28,43)(25,39,41,40)(29,44,45,46)(37,47,42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,e^a=eh,g^a=ghi,c^b=ch,e^b=ei,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90101);