#197935

order	512
nilpotency class	3
isoclinism	isoclinic class of #19384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105131);