#242920

order	512
nilpotency class	3
isoclinism	isoclinic class of #238001
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#397
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

605 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150116);