#111491

order	512
nilpotency class	3
isoclinism	isoclinic class of #10564
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#3186	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

808 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18687);