#233601

order	512
nilpotency class	3
isoclinism	isoclinic class of #232809
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1203 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1203 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140797);