#239072

order	512
nilpotency class	3
isoclinism	isoclinic class of #233152
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

880 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146268);