#481024

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#2657	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#559 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,f^b=fgi,d^c=dhi,f^e=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388220);