#6345397

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
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	C₂²×C₄×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1562 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ehi,f^a=fh,f^b=fh,d^c=di,e^c=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252593);