#238996

order	512
nilpotency class	3
isoclinism	isoclinic class of #238990
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	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

830 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 182 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)(17,18,23,28)(19,29,32,41)(20,34,36,25)(21,27,22,26)(24,33,40,31)(30,45,46,42)(35,48,47,39)(37,43,38,44);(1,3)(2,7)(4,12)(5,11)(6,13)(8,15)(9,14)(10,16)(17,19,23,32)(18,24,28,40)(20,35,36,47)(21,33,22,31)(25,42,34,45)(26,41,27,29)(30,37,46,38)(39,43,48,44);(2,8)(3,11)(4,5)(9,10)(12,13)(14,15)(17,20,21,37,23,36,22,38)(18,25,26,43,28,34,27,44)(19,30,31,47,32,46,33,35)(24,39,29,45,40,48,41,42);(20,36)(25,34)(30,46)(35,47)(37,38)(39,48)(42,45)(43,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=e,e^2=h,f^2=i,g^2=hi,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=ghi,d^c=dh,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146192);