#497095

order	512
nilpotency class	4
isoclinism	isoclinic class of #33752
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=efh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce

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, 404291);