#113403

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

847 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 20599);