#6489325

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1561 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=ghi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396521);