#150654

order	512
nilpotency class	5
isoclinism	isoclinic class of #150594
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2311 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

300 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 52 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,21,26,22,27,37,42)(19,28,32,23,34,40,46,41)(20,24,36,25,29,38,35,43)(30,45,31,39,33,44,47,48);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,29,30,22,34,20,33)(18,23,38,48,27,41,24,39)(21,31,36,32,37,47,35,46)(25,28,42,45,43,40,26,44);(2,7)(6,14)(8,15)(9,16)(17,20,22,29)(18,24,27,38)(19,30,34,33)(21,36,37,35)(23,39,41,48)(25,42,43,26)(28,45,40,44)(31,46,47,32)

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=i,d^2=gh,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dg,e^b=efghi,d^c=di,e^d=ehi,f^e=fi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57850);