#7659041

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6342 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566237);