#215723

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

701 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)(17,18,21,26,22,25,20,24)(19,27,28,32,29,23,30,31)(33,34)(35,40)(36,45)(37,42)(38,47)(39,43)(41,46)(44,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(33,35,39,36)(34,40,43,45)(37,46,38,44)(41,47,48,42);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,22,29)(18,23,25,27)(20,28,21,30)(24,31,26,32)(33,36,39,35)(34,41,43,48)(37,46,38,44)(40,47,45,42);(2,8)(6,9)(7,16)(14,15)(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,27,32)(33,37,39,38)(34,42,43,47)(35,44,36,46)(40,48,45,41)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=eh,d^2=h,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, 122919);