#6460880

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1583 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,28,37)(20,30,33,47)(21,34,35,25)(24,39,41,32)(27,42,31,46)(29,45,44,38)(36,48,40,43);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,38)(26,37)(30,42)(31,33)(32,43)(34,45)(35,44)(39,48)(40,41)(46,47);(10,14)(11,15)(17,20,21,32,22,33,35,39)(18,24,25,30,26,41,34,47)(19,27,29,43,28,31,44,48)(23,36,38,42,37,40,45,46);(11,15)(13,16)(17,19)(18,23)(20,31)(21,29)(22,28)(24,40)(25,38)(26,37)(27,33)(30,46)(32,48)(34,45)(35,44)(36,41)(39,43)(42,47);(1,3,4,7)(2,6,5,8)(19,28)(23,37)(27,31)(29,44)(36,40)(38,45)(42,46)(43,48)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^d=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, 6368076);