#486016

order	512
nilpotency class	4
isoclinism	isoclinic class of #485617
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	#18322	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

790 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)(5,14)(6,9)(7,11)(12,16)(13,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,42)(29,43)(31,47)(32,39)(33,40)(34,48)(35,41)(36,44)(37,45)(38,46);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,16)(10,14)(17,19)(18,23)(20,31)(21,30)(22,29)(24,38)(25,27)(26,37)(28,33)(32,46)(34,44)(35,45)(36,40)(39,47)(41,43)(42,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,20,22,33)(18,24,26,40)(19,28,29,31)(21,34,35,32)(23,36,37,38)(25,39,41,48)(27,42,43,47)(30,46,45,44);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(19,29)(23,37)(27,43)(28,31)(30,45)(36,38)(42,47)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284971, #72285064, #72285473
maximal normal subgroup of	#72284971, #72285064, #72285473

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393212);