#221157

order	512
nilpotency class	3
isoclinism	isoclinic class of #221144
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3506	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

835 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)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(17,18,24,21)(19,28,31,41)(20,32)(22,37,23,27)(25,38,36,30)(26,39)(29,44)(33,34)(35,42)(40,47)(43,45)(46,48);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19,24,31)(18,25,21,36)(20,29,35,47)(22,38,23,30)(26,40,33,44)(27,41,37,28)(32,43,42,48)(34,46,39,45);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(17,20)(18,26)(19,29)(21,33)(22,34)(23,39)(24,35)(25,40)(27,42)(28,43)(30,45)(31,47)(32,37)(36,44)(38,46)(41,48);(17,21,24,18)(19,28,31,41)(20,33,35,26)(22,27,23,37)(25,38,36,30)(29,43,47,48)(32,34,42,39)(40,46,44,45)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=de,e^b=eh,f^c=fhi,g^c=gh,e^d=eh

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, 128353);