#122263

order	512
nilpotency class	4
isoclinism	isoclinic class of #11692
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
socle	C₂²	C₂×#323
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

332 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 84 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)(33,34,37,42,38,43,52,60,39,44,53,61,36,41,50,58)(35,45,47,62,46,56,63,59,48,57,51,64,49,40,54,55);(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)(36,49)(37,51)(38,46)(39,48)(41,57)(42,59)(43,45)(44,56)(47,53)(50,63)(52,54)(55,61)(58,62)(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)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=i,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=di,e^b=efi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29459);