#115608

order	512
nilpotency class	3
isoclinism	isoclinic class of #115600
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

744 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

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

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, 22804);