#332774

order	512
nilpotency class	3
isoclinism	isoclinic class of #332711
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

570 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)(3,11)(4,14)(5,15)(6,10)(7,8)(9,12)(13,16)(17,18,23,28)(19,29,32,37)(20,26,35,42)(21,24,22,27)(25,36,40,30)(31,45,46,43)(33,38,34,41)(39,44,48,47);(1,3,6,7)(2,8,10,11)(4,12,5,13)(9,14,16,15)(18,24)(19,30)(20,33)(21,22)(27,28)(29,37)(31,46)(32,36)(34,35)(38,41)(39,43)(45,48);(1,4,6,5)(2,9,10,16)(3,12,7,13)(8,14,11,15)(17,19,23,32)(18,25,28,40)(20,31,35,46)(21,30,22,36)(24,37,27,29)(26,39,42,48)(33,44,34,47)(38,45,41,43);(1,5,6,4)(2,9,10,16)(3,13,7,12)(8,14,11,15)(17,20)(18,26)(19,31)(21,33)(22,34)(23,35)(24,38)(25,39)(27,41)(28,42)(29,43)(30,44)(32,46)(36,47)(37,45)(40,48)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,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=dei,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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239970);