#182859

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

967 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,26,23,27,21,37)(19,28,32,39,30,42,31,24)(20,33,35,48,36,25,34,40)(29,43,46,38,44,41,45,47);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,19)(18,24,27,39)(20,29)(21,31)(22,32)(23,30)(25,38,33,47)(26,28,37,42)(34,45)(35,46)(36,44)(40,41,48,43);(17,20,22,35,23,36,21,34)(18,25,26,40,27,33,37,48)(19,29,32,46,30,44,31,45)(24,38,28,41,39,47,42,43);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,30)(24,39)(28,42)(29,44)(31,32)(38,47)(41,43)(45,46)

Polycyclic presentation:

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

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, 90055);