#6400728

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1743 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,d^b=df,e^b=ei,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6307924);