#120926

order	512
nilpotency class	3
isoclinism	isoclinic class of #120924
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 358 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)(5,10)(6,9)(8,17)(12,21,29,27)(13,28,31,22)(16,23,30,32)(18,26)(19,24)(20,25)(33,34,38,42,39,44,37,43)(35,45,47,40,49,56,48,57)(36,50,52,60,53,59,54,41)(46,58,63,64,51,55,62,61);(1,3)(2,7)(4,16,17,30)(5,13)(6,14)(8,23,15,32)(9,11)(10,22)(12,18,29,19)(20,31)(21,24,27,26)(25,28)(33,35,37,48,39,49,38,47)(34,40,43,45,44,57,42,56)(36,51,52,62,53,46,54,63)(41,58,50,64,60,55,59,61);(1,4)(2,8,9,15)(3,12)(5,19)(6,17)(7,21,11,27)(10,26,25,24)(13,30)(14,29)(16,31)(18,20)(22,32,28,23)(33,36,37,52,39,53,38,54)(34,41,42,59,44,60,43,50)(35,46,47,62,49,51,48,63)(40,55,45,61,57,58,56,64)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,g^b=ghi,e^c=ehi,f^c=fh,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, 28122);