#484941

order	512
nilpotency class	4
isoclinism	isoclinic class of #484587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#18306	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

785 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,27)(20,31)(21,34)(22,26)(23,30)(24,32)(25,37)(28,43)(29,41)(33,40)(35,39)(36,48)(38,46)(42,45)(44,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,23)(20,28)(21,35)(22,29)(24,38)(25,41)(26,39)(27,34)(30,37)(31,47)(32,45)(33,46)(36,44)(40,43)(42,48);(17,20)(18,24,26,31)(19,28)(21,36)(22,32)(23,38,39,47)(25,40,34,48)(27,42,41,43)(29,45)(30,46)(33,37)(35,44);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,29)(20,32)(23,39)(24,31)(27,41)(30,35)(33,36)(40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392137);