#219631

order	512
nilpotency class	3
isoclinism	isoclinic class of #219624
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

534 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=ef,d^2=ei,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dhi,f^b=fh,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, 126827);