#132145

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×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2312 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 489 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,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, 39341);