#251823

order	512
nilpotency class	3
isoclinism	isoclinic class of #251818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2404
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

726 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,d^b=dhi,f^b=fh,e^c=eh,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159019);