#123170

order	512
nilpotency class	4
isoclinism	isoclinic class of #123151
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

236 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 44 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30366);