#7715464

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 56 points:

(2,5)(6,8)(9,10)(11,15)(12,19)(13,21)(14,17)(16,22)(18,24)(20,23)(25,26,31,27)(28,32,38,35)(29,33,42,36)(30,37,43,34)(39,44,50,47)(40,48,51,45)(41,49,53,46)(52,55,56,54);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,15,17,19)(13,18,22,20)(16,23,21,24)(26,27)(28,38)(33,36)(34,37)(39,50)(40,51)(46,49)(52,56);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,15,17,19)(13,20,22,18)(16,23,21,24)(25,27,31,26)(28,35,38,32)(29,33,42,36)(30,37,43,34)(39,44,50,47)(40,48,51,45)(41,46,53,49)(52,54,56,55);(9,10)(11,15)(12,19)(13,16)(14,17)(18,23)(20,24)(21,22)(25,28)(26,32)(27,35)(29,39)(30,40)(31,38)(33,44)(34,45)(36,47)(37,48)(41,52)(42,50)(43,51)(46,54)(49,55)(53,56);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,29)(26,33)(27,36)(28,39)(30,41)(31,42)(32,44)(34,46)(35,47)(37,49)(38,50)(40,52)(43,53)(45,54)(48,55)(51,56);(25,30,31,43)(26,34,27,37)(28,40,38,51)(29,41,42,53)(32,45,35,48)(33,46,36,49)(39,52,50,56)(44,54,47,55)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,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, 7622660);