#7125414

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=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, 7032610);