#488374

order	512
nilpotency class	4
isoclinism	isoclinic class of #487659
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×C₂²⋊C₄	C₂⁴
prime radical	#18210	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

838 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,13)(4,9)(5,11)(6,10)(7,12)(8,14)(15,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,42)(30,43)(31,47)(32,39)(33,40)(34,48)(35,41)(36,44)(37,46)(38,45);(1,3,7,4)(2,8,12,16)(5,14,6,15)(9,10,13,11)(17,19,22,30)(18,23,26,37)(20,31,32,28)(21,29,35,46)(24,38,39,36)(25,27,41,43)(33,45,34,44)(40,47,48,42);(1,4,7,3)(2,9,12,13)(5,14,6,15)(8,10,16,11)(17,20)(18,24)(19,28)(21,34)(22,32)(23,36)(25,40)(26,39)(27,42)(29,45)(30,31)(33,35)(37,38)(41,48)(43,47)(44,46);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,16,9)(20,32)(24,39)(28,31)(33,34)(36,38)(40,48)(42,47)(44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395570);