#7715564

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×#3561	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6284 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)(5,6,11,9)(7,12,8,14)(10,17,18,13)(15,20,16,19)(21,22)(23,27)(24,28)(25,32)(26,33)(29,38)(30,39)(31,34)(35,43)(36,44)(37,42)(40,46)(41,47)(45,49)(48,50)(51,52);(1,2,3,4)(5,7,11,8)(6,12,9,14)(10,15,18,16)(13,19,17,20)(22,23)(24,34)(29,32)(30,33)(35,46)(36,47)(42,45)(48,52);(5,8,11,7)(6,12,9,14)(10,16,18,15)(13,19,17,20)(21,23,27,22)(24,28,34,31)(25,29,38,32)(26,33,39,30)(35,43,46,40)(36,41,47,44)(37,42,49,45)(48,51,52,50);(5,9,11,6)(7,14,8,12)(10,17,18,13)(15,20,16,19)(21,24,27,34)(22,28,23,31)(25,35,38,46)(26,36,39,47)(29,40,32,43)(30,41,33,44)(37,48,49,52)(42,50,45,51);(5,10,11,18)(6,13,9,17)(7,15,8,16)(12,19,14,20)(21,25,27,38)(22,29,23,32)(24,35,34,46)(26,37,39,49)(28,40,31,43)(30,42,33,45)(36,48,47,52)(41,50,44,51);(21,26,27,39)(22,30,23,33)(24,36,34,47)(25,37,38,49)(28,41,31,44)(29,42,32,45)(35,48,46,52)(40,50,43,51)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,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, 7622760);