#6677311

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6584507);