#149928

order	512
nilpotency class	5
isoclinism	isoclinic class of #149927
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#382	C₂³
prime radical	#2619	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2317 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

290 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 54 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)(17,18,21,26,22,28,38,41)(19,29,32,40,33,43,47,23)(20,24,36,42,37,27,35,25)(30,46,31,45,34,44,48,39);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,34)(21,31)(22,33)(24,39)(25,43)(26,44)(27,45)(28,40)(29,42)(30,37)(32,36)(35,47)(38,48)(41,46);(3,9)(6,12)(10,15)(13,16)(17,20,22,37)(18,24,28,27)(19,30,33,34)(21,36,38,35)(23,39,40,45)(25,26,42,41)(29,46,43,44)(31,47,48,32)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57124);