#236075

order	512
nilpotency class	3
isoclinism	isoclinic class of #234044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1220 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143271);