#480971

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	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#554 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

774 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)(19,27,29,42)(20,24)(21,34,35,25)(22,26)(23,30)(28,41,31,45)(32,39)(33,48,47,40)(36,43)(37,44)(38,46);(17,19,22,29)(18,23)(20,31,32,28)(21,30,35,44)(24,38)(25,27)(26,37)(33,46,47,43)(34,42)(36,39)(40,45)(41,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,36)(25,40)(26,39)(27,41)(29,31)(30,43)(34,48)(35,47)(37,38)(42,45)(44,46);(3,9)(6,12)(10,15)(13,16)(19,29)(20,32)(23,37)(24,39)(27,42)(30,44)(33,47)(40,48)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,d^b=di,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, 388167);