#182915

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	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

873 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=h,g^2=i,b^a=be,d^a=dh,e^a=eh,g^a=ghi,d^b=dh,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, 90111);