#149757

order	512
nilpotency class	5
isoclinism	isoclinic class of #149757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11501
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	#407	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2921 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 405 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,4,9,5,10,17,25,6,7,15,20,18,21,30,32)(3,11,13,26,14,19,29,24,12,27,16,31,28,8,22,23);(2,7)(4,15)(5,18)(8,19)(9,20)(13,16)(14,28)(23,24);(1,3)(2,8,7,19)(4,16)(5,14)(6,12)(9,24,20,23)(10,11,21,27)(13,15)(17,22)(18,28)(25,31,32,26)(29,30)

Polycyclic presentation:

2^9|a^2=e,c^2=h,e^2=g,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,e^b=ef,g^b=gi,d^c=di,e^c=ei,e^d=eh,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, 56953);