#482334

order	512
nilpotency class	4
isoclinism	isoclinic class of #482080
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

735 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)(4,8)(5,7)(10,15,16,13)(12,14)(18,24,26,38)(20,33)(21,22)(25,37,40,47)(27,34,28,39)(30,44)(31,32)(35,36)(41,45,42,48)(43,46);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,18,23,26)(19,25,29,40)(20,34,36,39)(21,28,22,27)(24,33,38,35)(30,45,43,48)(31,42,32,41)(37,44,47,46);(2,7)(6,9)(8,14)(13,15)(17,19)(18,25)(20,30)(21,31)(22,32)(23,29)(24,37)(26,40)(27,41)(28,42)(33,44)(34,45)(35,46)(36,43)(38,47)(39,48);(1,4,5,12)(2,8,7,14)(3,10,11,16)(6,13,9,15)(19,29)(25,40)(30,43)(31,32)(37,47)(41,42)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=gh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389530);