#112100

order	512
nilpotency class	3
isoclinism	isoclinic class of #100051
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19296);