#263349

order	512
nilpotency class	3
isoclinism	isoclinic class of #263331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	C₂×#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 760 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=i,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,d^c=di,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170545);