#244145

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
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	C₂³	#525
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

579 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=ef,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,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151341);