#484891

order	512
nilpotency class	4
isoclinism	isoclinic class of #12155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
socle	C₂²	C₂×B(4,𝔽₂)
prime radical	#2006	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

739 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 739 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,24)(20,32)(21,26)(22,36)(23,28)(25,34)(27,38)(29,40)(30,46)(31,42)(33,47)(35,44)(37,41)(39,45)(43,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,23,31)(18,24,28,42)(20,33,21,29)(22,37,38,30)(25,43,44,39)(26,40,32,47)(27,41,36,46)(34,45,35,48);(17,20,23,21)(18,25)(19,29,31,33)(22,34,38,35)(24,39)(26,36)(27,32)(28,44)(30,45,37,48)(40,41)(42,43)(46,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,20)(18,26,28,32)(19,29,31,33)(22,35,38,34)(24,40,42,47)(25,36,44,27)(30,45,37,48)(39,41,43,46)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,e^c=eh,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392087);