#7714954

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

6275 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)(25,37)(30,39)(33,44)(35,48)(41,49)(42,47)(46,51)(50,52);(1,2)(3,4)(22,27)(23,31)(29,34)(30,39)(32,43)(33,44)(42,47)(46,51);(6,10)(11,13)(12,17)(18,19)(21,22)(23,28)(24,34)(25,30)(26,27)(29,36)(31,38)(32,45)(33,41)(35,47)(37,39)(40,43)(42,48)(44,49)(46,52)(50,51);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,23,26,31)(22,28,27,38)(24,32,36,43)(25,33,37,44)(29,40,34,45)(30,41,39,49)(35,46,48,51)(42,50,47,52);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,24,26,36)(22,29,27,34)(23,32,31,43)(25,35,37,48)(28,40,38,45)(30,42,39,47)(33,46,44,51)(41,50,49,52);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,25)(22,30)(23,33)(24,35)(26,37)(27,39)(28,41)(29,42)(31,44)(32,46)(34,47)(36,48)(38,49)(40,50)(43,51)(45,52)

Polycyclic presentation:

2^9|d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,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, 7622150);