#484755

order	512
nilpotency class	4
isoclinism	isoclinic class of #484601
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

722 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=di,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, 391951);