#7034618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1789 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,28,40,27,25,26,31)(23,32,41,29,35,45,34,38)(24,30,42,52,36,39,37,44)(33,46,50,43,48,47,49,51);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,35)(22,29,25,38)(24,33,36,48)(26,41,28,34)(30,43,39,51)(31,45,40,32)(37,50,42,49)(44,47,52,46);(2,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,36)(22,30,25,39)(23,33,35,48)(26,37,28,42)(29,43,38,51)(31,44,40,52)(32,46,45,47)(34,49,41,50);(8,16)(11,20)(13,15)(17,19)(21,25,28,31,27,22,26,40)(23,32,41,29,35,45,34,38)(24,30,42,52,36,39,37,44)(33,47,50,51,48,46,49,43);(6,12)(7,14)(11,20)(13,15)(24,36)(30,39)(33,48)(37,42)(43,51)(44,52)(46,47)(49,50)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=i,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6941814);