#7714919

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

6276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

(2,4)(6,7)(9,17)(11,13)(15,20)(25,37)(30,39)(33,44)(35,48)(41,49)(42,47)(46,51)(50,52);(1,2)(3,4)(5,6,10,7)(8,11,16,13)(9,12,17,14)(15,18,20,19)(22,27)(23,31)(29,34)(30,39)(32,43)(33,44)(42,47)(46,51);(5,7,10,6)(8,11,16,13)(9,14,17,12)(15,18,20,19)(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);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(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);(5,9)(6,12)(7,14)(8,15)(10,17)(11,18)(13,19)(16,20)(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);(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|b^2=h,c^2=h,d^2=hi,e^2=i,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, 7622115);