#149925

order	512
nilpotency class	5
isoclinism	isoclinic class of #149923
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#382	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#382 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

271 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=h,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cgi,d^b=dhi,e^b=efhi,f^b=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57121);