#221229

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	#61217	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

888 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)(3,6)(4,7)(5,9)(8,12)(10,13)(11,15)(14,16)(17,18,22,28,24,20,23,27)(19,29,33,30,34,41,32,25)(21,26,38,43,39,35,37,44)(31,45,36,46,48,42,47,40);(1,3,5,11)(2,6,9,15)(4,10,12,16)(7,13,8,14)(17,19)(18,25)(20,30)(21,36)(22,33)(23,32)(24,34)(26,42)(27,41)(28,29)(31,37)(35,45)(38,48)(39,47)(40,43)(44,46);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(17,20,24,18)(19,30,34,25)(21,35,39,26)(22,28,23,27)(29,32,41,33)(31,46,48,40)(36,45,47,42)(37,44,38,43);(17,21)(18,26)(19,31)(20,35)(22,38)(23,37)(24,39)(25,40)(27,44)(28,43)(29,45)(30,46)(32,47)(33,36)(34,48)(41,42)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,e^2=h,f^2=hi,g^2=h,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, 128425);