#485376

order	512
nilpotency class	4
isoclinism	isoclinic class of #485337
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	#534	C₂³
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 721 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,29)(20,27)(21,34)(22,25)(23,28)(24,30)(26,36)(31,45)(32,42)(33,40)(35,41)(37,39)(38,46)(43,47)(44,48);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19)(18,24)(20,30)(21,31)(22,37)(23,32)(25,29)(26,38)(27,42)(28,39)(33,44)(34,48)(35,46)(36,47)(40,43)(41,45);(17,20)(18,25,28,27)(19,30)(21,35)(22,23)(24,29,39,42)(26,41,34,40)(31,46)(32,37)(33,36)(38,45,48,43)(44,47);(2,7)(6,15)(8,12)(9,16)(17,21,23,36)(18,26,28,34)(19,31,32,47)(20,33,22,35)(24,38,39,48)(25,40,27,41)(29,43,42,45)(30,44,37,46)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392572);