#6852624

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6759820);