#152196

order	512
nilpotency class	5
isoclinism	isoclinic class of #152194
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12759
commutator	C₄⋊Q₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊Q₈	C₂³
prime radical	#61111	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

804 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dhi,e^b=eghi,f^b=fgh,g^b=gi,d^c=dgi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ehi,f^d=fhi,g^e=gi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59392);