#134698

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

269 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41894);