#152115

order	512
nilpotency class	5
isoclinism	isoclinic class of #152115
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12696
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#60823	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

650 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,14,17,8)(4,19,21,37)(5,11)(6,12,28,10)(7,13)(9,22,34,24)(15,42,44,59)(16,41,47,30)(18,31,27,32)(20,51,45,49)(23,50)(25,38,56,36)(26,39)(29,43,57,64)(33,54)(35,53)(40,63,58,46)(48,60,55,61)(52,62);(1,3)(2,8)(4,20)(5,16)(6,27)(7,18)(9,29)(10,31)(11,30)(12,14)(13,32)(15,21)(17,28)(19,49)(22,46)(23,45)(24,43)(25,55)(26,48)(33,40)(34,58)(35,57)(36,60)(37,59)(38,41)(39,61)(42,50)(44,52)(47,56)(51,62)(53,63)(54,64);(1,4)(2,9)(3,15)(5,24)(6,23)(7,21)(8,29)(10,35)(11,37)(12,33)(13,34)(14,40)(16,46)(17,45)(18,20)(19,39)(22,26)(25,54)(27,44)(28,52)(30,59)(31,57)(32,58)(36,50)(38,62)(41,51)(42,60)(43,48)(47,64)(49,61)(53,56)(55,63)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59311);