#219299

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cei,d^b=dh,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126495);