#6344441

order	512
nilpotency class	3
isoclinism	isoclinic class of #61591
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₄⋊C₄
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1551 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,e^b=ei,d^c=dhi

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, 6251637);