#332618

order	512
nilpotency class	3
isoclinism	isoclinic class of #332583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

595 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)(3,10)(4,6)(5,9)(7,11)(8,13)(12,16)(14,15)(17,18,23,28)(19,29,33,40)(20,34,36,26)(21,24,22,27)(25,30,41,32)(31,43,46,48)(35,38,37,42)(39,44,47,45);(18,24)(19,30)(20,35)(21,22)(26,34)(27,28)(29,40)(32,33)(36,37)(39,43)(44,45)(47,48);(2,6)(4,13)(7,10)(8,9)(11,14)(15,16)(17,19)(18,25)(20,31)(21,30)(22,32)(23,33)(24,29)(26,39)(27,40)(28,41)(34,47)(35,45)(36,46)(37,44)(38,43)(42,48);(1,3,5,12)(2,7,9,15)(4,14,13,11)(6,10,8,16)(17,20)(18,26)(19,31)(21,37)(22,35)(23,36)(24,38)(25,39)(27,42)(28,34)(29,43)(30,44)(32,45)(33,46)(40,48)(41,47)

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gh,f^c=fi,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, 239814);