#211843

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂³	#525
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

545 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)(3,8)(4,13)(5,10)(6,15)(7,11)(9,14)(12,16)(17,18,22,26,23,27,21,37)(19,28,32,38,29,43,31,24)(20,25,35,41,36,42,34,48)(30,44,46,40,33,47,45,39);(1,3,7,4)(2,8,11,13)(5,14,6,12)(9,15,16,10)(17,19,21,31,23,29,22,32)(18,24,26,28,27,38,37,43)(20,33,34,46,36,30,35,45)(25,40,41,47,42,39,48,44);(1,4,7,3)(2,9,11,16)(5,14,6,12)(8,15,13,10)(19,29)(24,38)(28,43)(30,33)(31,32)(39,40)(44,47)(45,46);(1,5,7,6)(2,10,11,15)(3,12,4,14)(8,16,13,9)(17,20,22,35,23,36,21,34)(18,25,26,41,27,42,37,48)(19,30,32,46,29,33,31,45)(24,39,28,44,38,40,43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=i,d^2=gi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,f^c=fi

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, 119039);