#6345849

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2922 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

(1,2,5,9,6,3,7,11)(4,8,13,15,12,10,14,16)(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);(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);(1,3,5,11,6,2,7,9)(4,10,13,16,12,8,14,15)(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);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(18,25)(20,32)(23,38)(26,34)(27,40)(28,43)(33,47)(44,48);(4,12)(8,10)(13,14)(15,16)

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,f^a=fh,f^b=fh,d^c=dh,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253045);