#132174

order	512
nilpotency class	4
isoclinism	isoclinic class of #131997
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	#51754	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 622 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,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, 39370);