#7992772

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7899968);