#133028

order	512
nilpotency class	4
isoclinism	isoclinic class of #132573
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄⋊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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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=egh,f^b=fgh,d^c=dghi,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, 40224);