#6669859

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1783 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6577055);