#238881

order	512
nilpotency class	3
isoclinism	isoclinic class of #238875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

860 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 188 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)(3,7,13,16)(4,8,5,9)(11,14,12,15)(17,18)(19,27)(20,31)(21,35)(22,26)(23,29)(24,33)(25,37)(28,42)(30,41)(32,46)(34,40)(36,39)(38,43)(44,47)(45,48);(1,3)(2,7)(4,12)(5,11)(6,13)(8,15)(9,14)(10,16)(17,19,22,30)(18,23,26,39)(20,32,33,28)(21,36,37,29)(24,38,31,44)(25,41,35,27)(34,43,48,47)(40,46,45,42);(2,8)(3,11)(4,5)(9,10)(12,13)(14,15)(17,20,21,34,22,33,37,48)(18,24,25,40,26,31,35,45)(19,28,29,43,30,32,36,47)(23,38,27,42,39,44,41,46);(20,33)(24,31)(28,32)(34,48)(38,44)(40,45)(42,46)(43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146077);