#122567

order	512
nilpotency class	4
isoclinism	isoclinic class of #122564
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

317 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,11)(4,8)(5,18,19,9)(6,10)(7,13,21,28)(12,25)(14,26)(15,29)(16,31,32,23)(17,24)(20,27,22,30)(33,34)(35,44,45,60)(36,49,51,64)(37,42)(38,54)(39,43)(40,46)(41,52,58,53)(47,57,48,63)(50,59)(55,61)(56,62);(1,3,6,14)(2,7,10,21)(4,15,17,12)(5,13,19,28)(8,22,24,20)(9,11,18,26)(16,30,32,27)(23,29,31,25)(33,35)(34,40)(36,50)(37,48)(38,47)(39,45)(41,57)(42,59)(43,55)(44,49)(46,53)(51,62)(52,61)(54,56)(58,63)(60,64);(1,4,6,17)(2,8,10,24)(3,12,14,15)(5,16,19,32)(7,20,21,22)(9,23,18,31)(11,25,26,29)(13,27,28,30)(35,45)(40,55)(44,60)(46,61)(47,48)(50,62)(56,59)(57,63)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=gi,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=chi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29763);