#7743450

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

6166 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)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(22,27)(23,31)(24,34)(30,39)(33,44)(36,47)(40,45)(50,52);(1,2,3,4)(22,27)(25,37)(28,38)(29,35)(33,44)(36,47)(40,45)(46,51);(6,10)(11,13)(12,17)(18,19)(21,22,26,27)(23,28,31,38)(24,35,34,29)(25,30,37,39)(32,45,43,40)(33,41,44,49)(36,48,47,42)(46,52,51,50);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,23,26,31)(22,28,27,38)(24,32,34,43)(25,33,37,44)(29,40,35,45)(30,41,39,49)(36,46,47,51)(42,50,48,52);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,24)(22,29)(23,32)(25,36)(26,34)(27,35)(28,40)(30,42)(31,43)(33,46)(37,47)(38,45)(39,48)(41,50)(44,51)(49,52);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,25,26,37)(22,30,27,39)(23,33,31,44)(24,36,34,47)(28,41,38,49)(29,42,35,48)(32,46,43,51)(40,50,45,52)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,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, 7650646);