#110109

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

556 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,13,16,8)(4,17,18,26)(5,11)(6,12)(7,10)(9,19,24,21)(14,28)(15,25,20,23)(22,30)(27,31)(29,32)(33,34,38,43,39,44,37,42)(35,45,49,40,48,55,47,54)(36,41,50,56,51,57,52,58)(46,60,61,64,62,59,63,53);(1,3,6,15,7,16,5,20)(2,8,12,23,10,13,11,25)(4,14,19,31,18,29,21,30)(9,22,17,28,24,27,26,32)(33,35)(34,40)(36,46)(37,49)(38,47)(39,48)(41,53)(42,55)(43,45)(44,54)(50,63)(51,62)(52,61)(56,59)(57,64)(58,60);(1,4,5,21,7,18,6,19)(2,9,11,26,10,24,12,17)(3,14,20,30,16,29,15,31)(8,22,25,32,13,27,23,28)(33,36,38,50,39,51,37,52)(34,41,43,56,44,57,42,58)(35,46,47,63,48,62,49,61)(40,53,45,59,54,64,55,60)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17305);