#240506

order	512
nilpotency class	3
isoclinism	isoclinic class of #233184
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

910 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)(7,10)(8,16)(9,13)(11,15)(12,18)(14,19)(17,20)(21,22,27,32)(23,33,35,45)(24,38,41,29)(25,31,26,30)(28,37,44,36)(34,49,50,46)(39,52,51,43)(40,47,42,48);(1,2)(3,4)(5,7)(6,10)(8,15)(9,14)(11,16)(12,20)(13,19)(17,18)(21,23,27,35)(22,28,32,44)(24,39,41,51)(25,37,26,36)(29,46,38,49)(30,45,31,33)(34,40,50,42)(43,47,52,48);(2,4)(5,7)(6,11)(8,17)(9,14)(10,16)(12,19)(13,20)(15,18)(21,24,25,40,27,41,26,42)(22,29,30,47,32,38,31,48)(23,34,36,51,35,50,37,39)(28,43,33,49,44,52,45,46);(7,14)(10,19)(11,20)(15,17)(23,35)(28,44)(33,45)(34,50)(36,37)(39,51)(43,52)(46,49)

Polycyclic presentation:

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

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, 147702);