#132134

order	512
nilpotency class	4
isoclinism	isoclinic class of #132019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2313 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 470 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,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^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, 39330);