#7714842

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
prime radical	#47212	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6174 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)(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(22,27)(24,35)(25,38)(28,32)(33,43)(34,44)(42,49)(50,52);(1,2,3,4)(22,27)(23,31)(29,36)(30,39)(33,43)(34,44)(42,49)(47,51);(6,10)(11,13)(12,17)(18,19)(21,22,26,27)(23,32,31,28)(24,36,35,29)(25,30,38,39)(33,40,43,45)(34,46,44,41)(37,49,48,42)(47,50,51,52);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,23)(22,28)(24,33)(25,34)(26,31)(27,32)(29,40)(30,41)(35,43)(36,45)(37,47)(38,44)(39,46)(42,50)(48,51)(49,52);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,24)(22,29)(23,33)(25,37)(26,35)(27,36)(28,40)(30,42)(31,43)(32,45)(34,47)(38,48)(39,49)(41,50)(44,51)(46,52);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,25,26,38)(22,30,27,39)(23,34,31,44)(24,37,35,48)(28,41,32,46)(29,42,36,49)(33,47,43,51)(40,50,45,52)

Polycyclic presentation:

2^9|b^2=g,c^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, 7622038);