#220583

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	#525	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

818 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)(4,12,14,8)(5,9)(6,10)(11,16,13,15)(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);(1,3,6,5)(2,7,10,9)(4,11,14,13)(8,15,12,16)(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)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,20,24,35)(18,26,21,33)(19,29,31,47)(22,34,23,39)(25,40,36,44)(27,42,37,32)(28,43,41,48)(30,45,38,46);(1,5,6,3)(2,9,10,7)(4,13,14,11)(8,16,12,15)(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=h,d^2=hi,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, 127779);