#489227

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61409	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396423);