#7743466

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

6258 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)(5,6,11,8)(7,13,9,12)(10,14,18,16)(15,20,17,19)(23,31)(28,32)(33,43)(34,44)(40,45)(41,46)(47,51)(50,52);(1,2)(3,4)(5,7,11,9)(6,12,8,13)(10,15,18,17)(14,19,16,20)(22,27)(25,38)(28,32)(29,35)(34,44)(36,49)(40,45)(47,51);(6,8)(12,13)(14,16)(19,20)(21,22,26,27)(23,32,31,28)(24,35,37,29)(25,30,38,39)(33,40,43,45)(34,46,44,41)(36,48,49,42)(47,50,51,52);(5,8,11,6)(7,13,9,12)(10,16,18,14)(15,20,17,19)(21,23,26,31)(22,28,27,32)(24,33,37,43)(25,34,38,44)(29,40,35,45)(30,41,39,46)(36,47,49,51)(42,50,48,52);(5,9,11,7)(6,13,8,12)(10,17,18,15)(14,20,16,19)(21,24,26,37)(22,29,27,35)(23,33,31,43)(25,36,38,49)(28,40,32,45)(30,42,39,48)(34,47,44,51)(41,50,46,52);(5,10,11,18)(6,14,8,16)(7,15,9,17)(12,19,13,20)(21,25)(22,30)(23,34)(24,36)(26,38)(27,39)(28,41)(29,42)(31,44)(32,46)(33,47)(35,48)(37,49)(40,50)(43,51)(45,52)

Polycyclic presentation:

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

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, 7650662);