#334810

order	512
nilpotency class	3
isoclinism	isoclinic class of #334695
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

619 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gi,f^c=fi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242006);