#480193

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

749 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,26,23,28,35,44,24,29,20,34,21,27,36,43)(19,30,31,47,33,41,48,40,32,45,38,46,37,25,39,42);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,25)(20,31)(21,37)(22,38)(23,33)(24,32)(26,40)(27,45)(28,30)(29,41)(34,42)(35,39)(36,48)(43,47)(44,46);(17,20,23,36,24,22,21,35)(18,26,28,44,29,34,27,43)(19,31,33,48,32,38,37,39)(25,40,30,46,41,42,45,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,32)(25,41)(30,45)(31,38)(33,37)(39,48)(40,42)(46,47)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=g,e^2=i,f^2=h,h^2=i,b^a=be,c^a=ci,e^a=eg,d^b=di,e^b=ei,f^b=fgi

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, 387389);