#7715559

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	C₂×#3554	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6268 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)(5,6,9,12)(7,10,15,13)(8,16,17,11)(14,19,20,18)(21,22)(23,25)(24,30)(26,40)(27,28)(29,31)(32,41)(33,37)(34,44)(35,42)(36,43)(38,51)(39,46)(45,48)(47,52)(49,50);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,17)(19,20)(22,28)(23,31)(24,35)(32,40)(34,46)(37,43)(38,48)(50,52);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,17)(18,20)(21,23)(22,29)(24,36)(25,28)(26,34)(27,31)(30,37)(32,44)(33,35)(38,49)(39,40)(41,46)(42,43)(45,50)(47,48)(51,52);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(21,24,27,35)(22,30,28,42)(23,33,31,36)(25,37,29,43)(26,38,41,48)(32,45,40,51)(34,47,46,49)(39,50,44,52);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,27,29)(22,31,28,23)(24,37,35,43)(26,39,41,44)(30,36,42,33)(32,46,40,34)(38,50,48,52)(45,49,51,47);(21,26,27,41)(22,32,28,40)(23,34,31,46)(24,38,35,48)(25,39,29,44)(30,45,42,51)(33,47,36,49)(37,50,43,52)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,e^2=h,f^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, 7622755);