#251772

order	512
nilpotency class	3
isoclinism	isoclinic class of #251761
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

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

normal subgroups only

This group has 733 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^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,d^c=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, 158968);