#221182

order	512
nilpotency class	3
isoclinism	isoclinic class of #221164
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	#61407	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

861 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)(17,18,23,27)(19,28,32,30)(20,33)(21,36,22,26)(24,37,40,31)(25,38)(29,44)(34,42)(35,41)(39,48)(43,46)(45,47);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19,23,32)(18,24,27,40)(20,29,35,48)(21,37,22,31)(25,39,42,44)(26,30,36,28)(33,43,41,45)(34,47,38,46);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(17,20)(18,25)(19,29)(21,34)(22,38)(23,35)(24,39)(26,41)(27,42)(28,43)(30,45)(31,46)(32,48)(33,36)(37,47)(40,44);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,18)(19,30)(20,25)(21,36)(22,26)(23,27)(24,37)(28,32)(29,45)(31,40)(33,34)(35,42)(38,41)(39,47)(43,48)(44,46)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dh,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128378);