#485136

order	512
nilpotency class	4
isoclinism	isoclinic class of #485065
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

724 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 116 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)(19,27)(20,31)(21,35)(22,26)(23,29)(24,33)(25,37)(28,44)(30,42)(32,43)(34,41)(36,39)(38,47)(40,45)(46,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,30)(18,23,26,39)(20,32,33,28)(21,36,37,29)(24,40,31,38)(25,42,35,27)(34,45,48,47)(41,43,46,44);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(20,33)(21,37)(24,31)(28,32)(29,36)(38,40)(41,46)(43,44);(2,9)(7,16)(8,13)(10,15)(17,20,22,33)(18,24,26,31)(19,28,30,32)(21,34,37,48)(23,38,39,40)(25,41,35,46)(27,43,42,44)(29,45,36,47)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=i,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=di,e^c=eh,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, 392332);