#238968

order	512
nilpotency class	3
isoclinism	isoclinic class of #238962
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

801 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 174 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)(3,4)(5,9)(6,10)(7,11)(8,15)(12,14)(13,16)(17,18,23,26)(19,28,32,43)(20,33,35,25)(21,38,22,27)(24,31,40,30)(29,44,45,41)(34,39,46,48)(36,42,37,47);(1,3)(2,4)(5,12)(6,8)(7,13)(9,14)(10,15)(11,16)(17,19,23,32)(18,24,26,40)(20,34,35,46)(21,31,22,30)(25,41,33,44)(27,43,38,28)(29,36,45,37)(39,47,48,42);(1,4)(2,8)(3,9)(5,14)(6,15)(7,16)(10,13)(11,12)(17,20,21,36,23,35,22,37)(18,25,27,42,26,33,38,47)(19,29,30,46,32,45,31,34)(24,39,28,41,40,48,43,44);(1,5,7,6)(2,9,11,10)(3,12,13,8)(4,14,16,15)(18,26)(20,35)(24,40)(27,38)(28,43)(29,45)(34,46)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=cg,e^b=eh,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, 146164);