#480889

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
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×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#555 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

838 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=ei,f^b=fgi,d^c=dh,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, 388085);