#332694

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	C₂³	#534
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

545 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,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=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, 239890);