#8057955

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5993 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965151);