#150748

order	512
nilpotency class	5
isoclinism	isoclinic class of #150397
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#390	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 ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 313 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,28,31,40)(20,24,33,43)(21,26,37,25)(22,27)(23,30,39,35)(29,45,32,38)(34,42)(36,44)(41,47,48,46);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,23,27,40)(20,32)(21,35)(22,30)(24,41,43,45)(25,28,26,39)(29,34)(31,37)(33,47)(36,46)(38,44,48,42);(2,7)(6,14)(8,15)(9,16)(17,20,22,33)(18,24,27,43)(19,29,30,46)(21,36,37,34)(23,38,40,48)(25,42,26,44)(28,45,39,41)(31,47,35,32)

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=i,d^2=gi,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=efg,f^b=fi,d^c=dh,e^c=ei,f^c=fi,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, 57944);