#480973

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 ◁ C₂×C₄⋊C₄ ◁ C₂×#556 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

756 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)(19,24)(20,25)(21,26,34,23)(22,27)(28,39)(29,38)(30,37,43,35)(31,42)(32,41)(33,40,45,36)(44,48,46,47);(18,23,27,26)(19,28)(20,31)(24,37,39,35)(25,40,42,36)(30,43)(33,45)(38,47,41,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,28)(18,24,27,39)(20,32,31,29)(21,30,34,43)(23,35,26,37)(25,41,42,38)(33,46,45,44)(36,48,40,47);(3,9)(6,12)(10,15)(13,16)(17,20,22,31)(18,25,27,42)(19,29,28,32)(21,33,34,45)(23,36,26,40)(24,38,39,41)(30,44,43,46)(35,47,37,48)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,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,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, 388169);