#239269

order	512
nilpotency class	3
isoclinism	isoclinic class of #238015
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 836 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:

(1,2,7,3)(4,12,13,16)(5,10,6,9)(8,15,11,14)(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);(1,3,7,2)(4,12,13,16)(5,9,6,10)(8,15,11,14)(17,19)(18,23)(20,32)(21,36)(22,30)(24,38)(25,41)(26,39)(27,35)(28,33)(29,37)(31,44)(34,43)(40,46)(42,45)(47,48);(1,4)(2,8)(3,11)(5,15)(6,14)(7,13)(9,16)(10,12)(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);(4,13)(8,11)(12,16)(14,15)(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=i,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=ghi,c^b=cg,e^b=eh,f^b=fhi,g^b=gi,d^c=dhi,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, 146465);