#317185

order	512
nilpotency class	3
isoclinism	isoclinic class of #317171
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

500 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

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

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, 224381);