#483448

order	512
nilpotency class	4
isoclinism	isoclinic class of #483439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂⁴	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=chi,d^b=di,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287136, #72287174, #72287216, …
maximal normal subgroup of	#72287136, #72287174, #72287216, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390644);