#149933

order	512
nilpotency class	5
isoclinism	isoclinic class of #149931
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	#384	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2317 ◁ G
LECS	G ▷ #384 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 266 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) = 128

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=hi,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cgh,d^b=dhi,e^b=efh,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, 57129);