#332682

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

575 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:

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

Polycyclic presentation:

2^9|a^2=hi,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, 239878);