#215889

order	512
nilpotency class	3
isoclinism	isoclinic class of #162662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=e,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ce,f^c=fi,g^c=ghi

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, 123085);