#120931

order	512
nilpotency class	3
isoclinism	isoclinic class of #120928
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	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

351 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,e^c=ehi,g^c=gi

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, 28127);