#132995

order	512
nilpotency class	4
isoclinism	isoclinic class of #132509
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

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

normal subgroups only

This group has 488 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,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, 40191);