#204299

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111495);