#314617

order	512
nilpotency class	3
isoclinism	isoclinic class of #251800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2482
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

499 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 128 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18,22,28,23,29,21,27)(19,24,31,40,32,26,38,45)(20,33,37,25,35,44,36,43)(30,41,34,42,46,48,47,39);(2,6)(7,15)(8,13)(9,16)(17,19)(18,24)(20,34,35,47)(21,38)(22,31)(23,32)(25,41,43,48)(26,29)(27,45)(28,40)(30,37,46,36)(33,39,44,42);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,20)(18,25)(19,30)(21,37)(22,36)(23,35)(24,39)(26,42)(27,44)(28,33)(29,43)(31,47)(32,46)(34,38)(40,48)(41,45);(2,6)(7,15)(8,13)(9,16)(17,19)(18,26)(20,30)(21,38)(22,31)(23,32)(24,29)(25,42)(27,40)(28,45)(33,41)(34,37)(35,46)(36,47)(39,43)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,f^b=fh,e^c=eh,f^c=fhi,g^c=gh,f^d=fh

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, 221813);