#220574

order	512
nilpotency class	3
isoclinism	isoclinic class of #220564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3489	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

919 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)(4,11,12,7)(5,8)(9,15,16,13)(10,14)(17,18)(19,28)(20,32,35,42)(21,24)(22,37)(23,27)(25,30)(26,34,33,39)(29,44,47,40)(31,41)(36,38)(43,46,48,45);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19)(18,25)(20,29)(21,36)(22,38)(23,30)(24,31)(26,40)(27,41)(28,37)(32,43)(33,44)(34,46)(35,47)(39,45)(42,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(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=i,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=de,e^b=eh,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127770);