#6685902

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1707 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)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22)(23,28)(24,35)(25,30)(26,40)(27,31)(29,38)(32,46)(33,42)(34,44)(36,45)(37,49)(39,43)(41,47)(48,51)(50,52);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,27,34)(22,28,31,44)(24,32,37,48)(25,39,26,33)(29,41,45,52)(30,43,40,42)(35,46,49,51)(36,50,38,47);(2,4)(21,24,26,38,27,37,25,36)(22,29,30,35,31,45,40,49)(23,32,33,47,34,48,39,50)(28,41,43,46,44,52,42,51);(6,11)(10,13)(12,19)(18,20)(21,25,27,26)(22,30,31,40)(23,33,34,39)(24,36,37,38)(28,42,44,43)(29,35,45,49)(32,47,48,50)(41,51,52,46);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(24,37)(29,45)(32,48)(35,49)(36,38)(41,52)(46,51)(47,50)

Polycyclic presentation:

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

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, 6593098);