#123209

order	512
nilpotency class	4
isoclinism	isoclinic class of #123191
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

226 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=fh,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=eh,e^d=ei

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30405);