#251735

order	512
nilpotency class	3
isoclinism	isoclinic class of #251731
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 731 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=di,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, 158931);