#111286

order	512
nilpotency class	3
isoclinism	isoclinic class of #10512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#60948	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 776 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)(3,11,13,7)(4,15,17,8)(5,10)(6,9)(12,29)(14,27,19,23)(16,32)(18,28,20,25)(21,26)(22,30)(24,31)(33,34,38,43,39,44,37,42)(35,45,48,40,49,55,47,56)(36,50,51,41,53,57,52,58)(46,60,61,64,62,54,63,59);(1,3)(2,7)(4,16)(5,19)(6,13)(8,24)(9,11)(10,27)(12,18)(14,21)(15,32)(17,31)(20,30)(22,25)(23,26)(28,29)(33,35)(34,40)(36,46)(37,48)(38,47)(39,49)(41,54)(42,55)(43,45)(44,56)(50,59)(51,63)(52,61)(53,62)(57,64)(58,60);(1,4)(2,8)(3,12)(5,20)(6,17)(7,22)(9,15)(10,28)(11,29)(13,30)(14,31)(16,19)(18,21)(23,32)(24,27)(25,26)(33,36)(34,41)(35,46)(37,51)(38,52)(39,53)(40,54)(42,57)(43,50)(44,58)(45,59)(47,61)(48,63)(49,62)(55,64)(56,60)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,e^c=ei,f^c=fh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18482);