#110247

order	512
nilpotency class	3
isoclinism	isoclinic class of #110242
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

529 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 529 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di,e^c=ehi,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17443);