#6376073

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1766 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)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22)(23,33)(24,38)(25,32)(26,31)(27,30)(28,36)(29,40)(34,49)(35,44)(37,45)(39,52)(41,48)(42,47)(43,51)(46,50);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23)(22,28)(24,39)(25,35)(26,36)(27,37)(29,46)(30,44)(31,33)(32,45)(34,41)(38,49)(40,51)(42,50)(43,47)(48,52);(7,14)(10,13)(15,20)(17,19)(21,24)(22,29)(23,34)(25,40)(26,41)(27,42)(28,43)(30,38)(31,47)(32,48)(33,46)(35,50)(36,39)(37,51)(44,52)(45,49);(6,11)(10,13)(12,18)(17,19)(21,25,26,27)(22,30,31,32)(23,35,36,37)(24,40,41,42)(28,44,33,45)(29,38,47,48)(34,50,39,51)(43,52,46,49);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(22,31)(23,36)(29,47)(30,32)(34,39)(35,37)(38,48)(50,51)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=cf,e^b=ei,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283269);