#483438

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,e^2=gi,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ch,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260357, #72260361, #72260411, …
maximal normal subgroup of	#72260357, #72260361, #72260411, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390634);