#105563

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#380
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ci,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12759);