#244154

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
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	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

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

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, 151350);