#116911

order	512
nilpotency class	3
isoclinism	isoclinic class of #116906
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

522 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

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

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, 24107);