#380767

order	512
nilpotency class	3
isoclinism	isoclinic class of #29748
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

990 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:

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

Polycyclic presentation:

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

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, 287963);