#116769

order	512
nilpotency class	3
isoclinism	isoclinic class of #116768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fi,e^c=ehi,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23965);