#194244

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1124 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101440);