#268268

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

858 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=hi,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fh,e^d=ei,f^d=fhi

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, 175464);