#141809

order	512
nilpotency class	4
isoclinism	isoclinic class of #141698
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33748	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

293 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,d^2=gi,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cgh,d^b=di,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210685, #72210688
maximal normal subgroup of	#72210685, #72210688
maximal quotient of	#72210531, #72210560, #72210562

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49005);