#112086

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

790 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)(5,6,11,13,10,7,9,15)(8,14,17,20,18,12,19,16)(21,22)(23,31,34,27)(24,28)(25,30)(26,29)(32,49,50,41)(33,47,38,42)(35,52,51,43)(36,45)(37,44)(39,48)(40,46);(5,7,10,6)(8,16,18,20)(9,13,11,15)(12,17,14,19)(21,23)(22,27)(24,35)(25,38)(26,34)(28,43)(29,31)(30,47)(32,36)(33,40)(37,51)(39,50)(41,45)(42,46)(44,52)(48,49);(2,4)(5,8,10,18)(6,12,7,14)(9,19,11,17)(13,16,15,20)(21,24,26,37)(22,28,29,44)(23,32,34,50)(25,39,40,36)(27,41,31,49)(30,48,46,45)(33,51,38,35)(42,52,47,43)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,f^2=h,g^2=i,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, 19282);