#215638

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
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, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

802 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)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30)(33,34,36,40,37,39,43,46)(35,38,42,45,41,44,47,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(33,35)(34,38)(36,42)(37,41)(39,44)(40,45)(43,47)(46,48);(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)(36,43)(38,44)(42,47);(2,8)(6,9)(7,16)(14,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(35,41)(38,44)(42,47)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122834);