#115848

order	512
nilpotency class	3
isoclinism	isoclinic class of #100003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

550 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fhi,f^c=fhi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23044);