#314629

order	512
nilpotency class	3
isoclinism	isoclinic class of #252235
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

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

normal subgroups only

This group has 504 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)(17,18,23,28,21,26,22,27)(19,29,32,44,31,24,38,40)(20,33,36,25,37,42,35,43)(30,46,34,39,47,41,48,45);(2,6)(7,15)(8,13)(9,16)(17,19,21,31)(18,24,26,29)(20,34)(22,38,23,32)(25,41)(27,44,28,40)(30,35)(33,39)(36,47)(37,48)(42,45)(43,46);(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)(26,43)(27,42)(28,33)(29,45)(31,47)(32,48)(34,38)(40,46)(41,44);(2,6)(7,15)(8,13)(9,16)(17,19)(18,24)(20,30)(21,31)(22,38)(23,32)(25,39)(26,29)(27,44)(28,40)(33,46)(34,36)(35,48)(37,47)(41,42)(43,45)

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=de,f^a=fi,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fhi,g^c=gh,f^d=fh

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, 221825);