#132966

order	512
nilpotency class	4
isoclinism	isoclinic class of #132521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#392	C₂×C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

492 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi

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, 40162);