#6345514

order	512
nilpotency class	3
isoclinism	isoclinic class of #61569
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	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1526 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252710);