#132962

order	512
nilpotency class	4
isoclinism	isoclinic class of #132529
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×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#392	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 493 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,f^c=fgi,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, 40158);