#112082

order	512
nilpotency class	3
isoclinism	isoclinic class of #99947
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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

770 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)(5,6,11,13,10,7,9,15)(8,14,17,20,18,12,19,16)(21,22,26,32,27,30,25,31)(23,33,36,43,37,28,35,44)(24,29,39,47,40,46,41,48)(34,49,38,52,50,42,51,45);(5,7,10,6)(8,16,18,20)(9,13,11,15)(12,17,14,19)(21,23,27,37)(22,28,30,33)(24,38,40,51)(25,36,26,35)(29,45,46,52)(31,44,32,43)(34,41,50,39)(42,48,49,47);(2,4)(5,8,10,18)(6,12,7,14)(9,19,11,17)(13,16,15,20)(21,24)(22,29)(23,34)(25,41)(26,39)(27,40)(28,42)(30,46)(31,48)(32,47)(33,49)(35,51)(36,38)(37,50)(43,52)(44,45)

Polycyclic presentation:

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

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, 19278);