#132095

order	512
nilpotency class	4
isoclinism	isoclinic class of #131979
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

611 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39291);