#6687303

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1749 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22,25,27)(23,31,32,47)(24,36,38,29)(26,40,33,30)(28,42,41,35)(34,49,37,46)(39,45,48,51)(43,50,44,52);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(22,27)(23,32)(29,36)(30,40)(34,37)(35,42)(45,51)(50,52);(2,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,23,33,42,25,32,26,35)(22,28,40,47,27,41,30,31)(24,37,48,50,38,34,39,52)(29,44,51,46,36,43,45,49);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,24,25,38)(22,29,27,36)(23,34,32,37)(26,39,33,48)(28,43,41,44)(30,45,40,51)(31,46,47,49)(35,50,42,52);(2,4)(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(24,38)(29,36)(34,37)(39,48)(43,44)(45,51)(46,49)(50,52)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594499);