#244518

order	512
nilpotency class	3
isoclinism	isoclinic class of #244499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61764	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

994 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)(4,7)(5,10)(6,9)(8,12)(13,16)(14,15)(17,18)(19,28)(20,23)(21,36)(22,35)(24,27)(25,30)(26,37)(29,40)(31,42)(32,33)(34,39)(38,41)(43,44)(45,47)(46,48);(2,7)(4,6)(8,11)(9,10)(12,14)(15,16)(17,19,23,31)(18,25,20,33)(21,29,38,45)(22,32,24,30)(26,40,34,47)(27,42,35,28)(36,43,41,48)(37,46,39,44);(1,2,5,10)(3,8,13,15)(4,7,6,9)(11,12,16,14)(17,20,23,18)(19,28,31,42)(21,34,38,26)(22,35,24,27)(25,30,33,32)(29,43,45,48)(36,37,41,39)(40,46,47,44);(1,3)(2,8)(4,14)(5,13)(6,12)(7,11)(9,16)(10,15)(17,21)(18,26)(19,29)(20,34)(22,39)(23,38)(24,37)(25,40)(27,41)(28,43)(30,46)(31,45)(32,44)(33,47)(35,36)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151714);