#7715529

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

6272 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)(6,10)(8,16)(11,13)(14,20)(21,22,26,27)(23,28,31,32)(24,35,37,29)(25,38,39,30)(33,45,43,40)(34,46,44,41)(36,42,49,48)(47,50,51,52);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(22,27)(23,31)(29,35)(30,38)(33,43)(34,44)(42,48)(47,51);(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22)(23,32)(24,35)(25,30)(26,27)(28,31)(29,37)(33,40)(34,46)(36,48)(38,39)(41,44)(42,49)(43,45)(47,50)(51,52);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,23)(22,28)(24,33)(25,34)(26,31)(27,32)(29,40)(30,41)(35,45)(36,47)(37,43)(38,46)(39,44)(42,50)(48,52)(49,51);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,24,26,37)(22,29,27,35)(23,33,31,43)(25,36,39,49)(28,40,32,45)(30,42,38,48)(34,47,44,51)(41,50,46,52);(21,25,26,39)(22,30,27,38)(23,34,31,44)(24,36,37,49)(28,41,32,46)(29,42,35,48)(33,47,43,51)(40,50,45,52)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,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, 7622725);