#298371

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

(2,3)(5,6,11,14)(7,12,16,18)(8,15,19,20)(9,13,10,17)(21,22)(23,32)(24,36)(25,30)(26,40)(27,31)(28,34)(29,39)(33,50)(35,44)(37,43)(38,48)(41,45)(42,47)(46,51)(49,52);(1,2,4,3)(5,7)(6,12)(8,17)(9,15)(10,20)(11,16)(13,19)(14,18)(21,23,25,34)(22,28,30,32)(24,37,38,52)(26,35,27,41)(29,46,47,50)(31,44,40,45)(33,42,51,39)(36,49,48,43);(5,8,10,18,11,19,9,12)(6,7,13,15,14,16,17,20)(21,24,26,39,25,38,27,42)(22,29,31,36,30,47,40,48)(23,33,35,52,34,51,41,37)(28,43,44,46,32,49,45,50);(1,3,4,2)(5,9,11,10)(6,13,14,17)(7,15,16,20)(8,12,19,18)(22,30)(24,38)(28,32)(31,40)(33,51)(37,52)(39,42)(44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205567);