#149827

order	512
nilpotency class	5
isoclinism	isoclinic class of #149800
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2129
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#423	C₂³
socle	C₂	#11515
prime radical	C₂⁴	#101
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2921 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 224 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) = 64
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=h,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,e^b=efhi,d^c=di,e^c=ei,e^d=ehi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57023);