#141808

order	512
nilpotency class	4
isoclinism	isoclinic class of #141701
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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	C₂³	#469
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

285 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 78 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)(19,27,30,41)(20,23)(21,33)(22,35)(24,31,36,47)(25,34)(26,32)(28,42,44,40)(29,46)(37,43)(38,45,39,48);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(18,23)(19,28,30,44)(21,29)(24,38,36,39)(26,35)(27,40,41,42)(31,48,47,45)(34,43);(3,11)(7,15)(10,16)(12,13)(17,19,29,47,22,28,43,48,20,30,21,31,32,44,34,45)(18,24,37,41,26,39,46,42,23,36,25,27,35,38,33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=ei,d^2=g,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,e^b=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287685, #72287693, #72287706
maximal normal subgroup of	#72287685, #72287693, #72287706

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49004);