#6570613

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6477809);