#213359

order	512
nilpotency class	3
isoclinism	isoclinic class of #19697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	#532	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

615 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,22,26,23,27,21,37)(19,28,31,39,32,44,30,24)(20,25,36,42,34,41,35,48)(29,43,46,40,33,47,45,38);(1,2)(3,11)(4,14)(5,8)(6,9)(7,13)(10,15)(12,16)(17,19,21,30,23,32,22,31)(18,24,26,28,27,39,37,44)(20,33,35,46,34,29,36,45)(25,40,42,47,41,38,48,43);(1,3,5,12,6,13,4,15)(2,7,8,10,9,11,14,16)(20,34)(25,41)(29,33)(35,36)(38,40)(42,48)(43,47)(45,46);(1,4,6,5)(2,8,9,14)(3,12,13,15)(7,16,11,10)(17,20,22,36,23,34,21,35)(18,25,26,42,27,41,37,48)(19,29,31,46,32,33,30,45)(24,38,28,43,39,40,44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,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=ci,d^b=dhi,d^c=dhi

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, 120555);