#484909

order	512
nilpotency class	4
isoclinism	isoclinic class of #484606
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×C₄⋊C₄	C₂⁴
prime radical	#12156	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

760 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)(20,24)(21,34)(22,26)(23,30)(25,35)(28,42)(29,41)(31,45)(32,47)(33,40)(36,43)(37,44)(38,46)(39,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(19,28)(23,36)(25,34)(29,31)(30,43)(37,38)(39,47)(44,46);(17,19)(18,23,26,36)(20,31)(21,30)(22,28)(24,38,40,37)(25,27,34,42)(29,33)(32,46)(35,43)(39,45,47,41)(44,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,33)(18,24,26,40)(19,29,28,31)(21,32,35,48)(23,37,36,38)(25,39,34,47)(27,41,42,45)(30,44,43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318727
maximal normal subgroup of	#72318727

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392105);