#149956

order	512
nilpotency class	5
isoclinism	isoclinic class of #149956
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11558
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	#407	C₂³
prime radical	#525	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 242 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,7,11,27,38,8,13,26,41,5,24,25,40)(3,14,17,43,16,9,29,31,18,33,46,56,28,32,53,39)(4,10,21,36,22,35,52,59,23,37,51,62,20,49,50,61)(15,42,44,60,19,30,47,54,45,34,57,58,48,55,64,63);(1,3)(2,9,13,32)(4,19)(5,16)(6,17)(7,28)(8,18)(10,34,37,42)(11,14,24,33)(12,39,41,31)(15,22)(20,45)(21,47)(23,48)(25,29)(26,46)(27,53)(30,49,55,35)(36,60,62,58)(38,56,40,43)(44,52)(50,57)(51,64)(54,59,63,61);(1,4,7,22,8,23,5,20)(2,10,11,35,13,37,24,49)(3,15,16,19,18,45,28,48)(6,21,27,52,26,51,25,50)(9,30,33,34,32,55,14,42)(12,36,38,59,41,62,40,61)(17,44,29,47,46,57,53,64)(31,54,56,58,39,63,43,60)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=g,d^2=i,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg,e^b=efi,g^b=gi,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, 57152);