#108506

order	512
nilpotency class	3
isoclinism	isoclinic class of #108504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#376
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

398 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,f^b=fi,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, 15702);