#120956

order	512
nilpotency class	3
isoclinism	isoclinic class of #120954
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

373 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 116 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)(3,11,14,7)(4,15,17,30)(5,10)(6,9)(8,16,21,31)(12,25)(13,24,28,20)(18,22)(19,26)(23,32)(27,29)(33,34,37,42,38,41,50,56)(35,39,45,53,44,52,48,57)(36,40,47,55,46,54,49,58)(43,51,60,62,59,63,61,64);(1,3,5,13,6,14,18,28)(2,7,10,20,9,11,22,24)(4,12,16,26,17,27,31,32)(8,19,15,29,21,23,30,25)(33,35)(34,39,41,52)(36,43,46,59)(37,48)(38,44)(40,51)(42,57,56,53)(45,50)(47,61,49,60)(54,63)(55,64)(58,62);(1,4)(2,8,9,21)(3,12)(5,16)(6,17)(7,19,11,23)(10,15,22,30)(13,26)(14,27)(18,31)(20,29,24,25)(28,32)(33,36)(34,40,41,54)(35,43,44,59)(37,49)(38,46)(39,51)(42,58,56,55)(45,61,48,60)(47,50)(52,63)(53,64)(57,62)

Polycyclic presentation:

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

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, 28152);