#141801

order	512
nilpotency class	4
isoclinism	isoclinic class of #141745
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

301 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225272, #72225275, #72225276
maximal normal subgroup of	#72225272, #72225275, #72225276

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48997);