#244140

order	512
nilpotency class	3
isoclinism	isoclinic class of #232760
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

616 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

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

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, 151336);