#112104

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

795 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)(7,13,16,10)(8,17)(9,11)(12,19)(14,15,20,18)(21,22,26,32,27,31,25,30)(23,33,36,28,37,43,35,42)(24,29,38,45,39,46,40,47)(34,48,49,41,50,52,51,44);(5,7)(6,10)(8,18)(9,16)(11,13)(12,14)(15,19)(17,20)(21,23,27,37)(22,28,31,42)(24,34,39,50)(25,36,26,35)(29,44,46,41)(30,43,32,33)(38,51,40,49)(45,52,47,48);(2,4)(5,6)(7,14)(8,17)(9,11)(10,18)(12,19)(13,15)(16,20)(21,24)(22,29)(23,34)(25,40)(26,38)(27,39)(28,41)(30,47)(31,46)(32,45)(33,48)(35,51)(36,49)(37,50)(42,44)(43,52)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=h,g^2=i,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, 19300);