#7186582

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=g,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=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, 7093778);