#113402

order	512
nilpotency class	3
isoclinism	isoclinic class of #97048
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#37606	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,d^c=dhi,e^c=eh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20598);