#482556

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#2617	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

916 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18)(19,27)(20,31,33,24)(21,34,35,25)(22,26)(23,30,36,42)(28,39)(29,43,41,40)(32,38)(37,44)(45,48)(46,47);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,37)(25,27)(26,36)(31,45)(32,44)(33,41)(34,39)(35,42)(38,40)(43,46)(47,48);(2,8)(7,15)(9,10)(11,16)(19,28)(23,36)(27,39)(29,41)(30,42)(37,45)(40,43)(44,48);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,20,22,33)(18,24,26,31)(19,29,28,41)(21,32,35,47)(23,37,36,45)(25,38,34,46)(27,40,39,43)(30,44,42,48)

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,f^d=fi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318670, #72318731, #72318754
maximal normal subgroup of	#72318670, #72318731, #72318754

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389752);