#7125577

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi

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, 7032773);