#194582

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1030 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)(7,13,14,20)(8,12)(9,11)(10,15,18,17)(16,19)(21,22,26,32,27,30,25,31)(23,33,37,28,35,43,36,44)(24,38,40,29,41,46,39,45)(34,42,50,51,48,52,49,47);(1,2,3,4)(8,16)(12,19)(13,20)(15,17)(21,23)(22,28)(24,34)(25,37)(26,36)(27,35)(29,42)(30,44)(31,43)(32,33)(38,51)(39,50)(40,49)(41,48)(45,52)(46,47);(5,7)(6,10)(8,17)(9,14)(11,18)(12,20)(13,19)(15,16)(21,24,27,41)(22,29,30,45)(23,34,35,48)(25,39,26,40)(28,42,44,52)(31,38,32,46)(33,47,43,51)(36,49,37,50);(6,11)(10,18)(12,19)(13,20)(22,30)(23,35)(29,45)(31,32)(34,48)(36,37)(38,46)(49,50)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dhi,f^a=fh,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gi

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, 101778);