#120986

order	512
nilpotency class	3
isoclinism	isoclinic class of #99283
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

402 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,g^b=ghi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28182);