#233738

order	512
nilpotency class	3
isoclinism	isoclinic class of #233734
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	#37541	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1224 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140934);