#183265

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
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×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

891 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)(3,10,12,7)(4,8,6,14)(11,16,13,15)(17,18,22,27,23,26,21,36)(19,28,32,38,30,42,31,24)(20,25,35,39,33,40,34,46)(29,41,44,47,45,48,43,37);(1,3,6,13,5,12,4,11)(2,7,8,15,9,10,14,16)(17,19)(18,24,26,38)(20,29)(21,31)(22,32)(23,30)(25,37,40,47)(27,28,36,42)(33,45)(34,43)(35,44)(39,41,46,48);(17,20,22,35,23,33,21,34)(18,25,27,39,26,40,36,46)(19,29,32,44,30,45,31,43)(24,37,28,41,38,47,42,48);(1,4,5,6)(2,8,9,14)(3,11,12,13)(7,15,10,16)(18,26)(19,30)(20,33)(27,36)(31,32)(34,35)(37,47)(41,48)

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,d^a=dh,e^a=eh,g^a=ghi,d^b=dh,e^b=ei,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90461);