#215717

order	512
nilpotency class	3
isoclinism	isoclinic class of #163726
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=eh,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122913);