#488389

order	512
nilpotency class	4
isoclinism	isoclinic class of #487643
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	#3496	C₂²
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

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

normal subgroups only

This group has 812 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) = 128

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,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, 395585);