#496431

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

Aliases

No aliases for this group.

Subgroups

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

774 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:

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

Polycyclic presentation:

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

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, 403627);