#6400604

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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×D₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1679 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22)(23,31)(24,35)(25,29)(26,40)(27,33)(28,38)(30,39)(32,48)(34,43)(36,52)(37,46)(41,45)(42,51)(44,49)(47,50);(5,7,10,17)(6,11,14,15)(8,16,9,18)(12,19,13,20)(21,23,25,33)(22,27,29,31)(24,36,37,32)(26,34,39,50)(28,44,45,42)(30,43,40,47)(35,48,46,52)(38,51,41,49);(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,25,37)(22,28,29,45)(23,32,33,36)(26,41,39,38)(27,42,31,44)(30,46,40,35)(34,51,50,49)(43,48,47,52);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(23,33)(27,31)(32,36)(34,50)(42,44)(43,47)(48,52)(49,51);(1,3,4,2)(6,14)(11,15)(12,13)(19,20)(22,29)(23,33)(24,37)(30,40)(34,50)(38,41)(42,44)(48,52)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6307800);