#332613

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	#28165	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

636 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)(3,8)(4,14)(5,15)(6,16)(7,11)(9,12)(10,13)(17,18)(19,30)(20,25)(21,28)(22,27)(23,29)(24,32)(26,36)(31,46)(33,42)(34,41)(35,45)(37,44)(38,43)(39,48)(40,47);(1,3,7,6)(2,8,11,16)(4,12,5,13)(9,15,10,14)(17,19,23,33)(18,24,29,41)(20,35,37,48)(21,34,22,32)(25,40,44,46)(26,45,43,39)(27,42,28,30)(31,38,47,36);(1,4,7,5)(2,9,11,10)(3,12,6,13)(8,15,16,14)(17,20)(18,25)(19,31)(21,36)(22,38)(23,37)(24,39)(26,28)(27,43)(29,44)(30,40)(32,35)(33,47)(34,48)(41,45)(42,46);(1,5,7,4)(2,10,11,9)(3,13,6,12)(8,14,16,15)(17,20)(18,26)(19,31)(21,38)(22,36)(23,37)(24,40)(25,28)(27,44)(29,43)(30,39)(32,48)(33,47)(34,35)(41,46)(42,45)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=i,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, 239809);