#110108

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 612 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(2,4)(5,6)(7,13,15,20)(8,12)(9,11)(10,14,18,16)(17,19)(21,22,25,31,27,30,26,32)(23,33,35,45,36,28,40,43)(24,37,39,44,38,46,41,29)(34,48,50,42,49,52,51,47);(5,7)(6,10)(8,16)(9,15)(11,18)(12,20)(13,19)(14,17)(21,23,26,35,27,36,25,40)(22,28,32,43,30,33,31,45)(24,34,39,51,38,49,41,50)(29,42,37,48,44,47,46,52);(1,2,3,4)(6,11)(8,17)(10,18)(14,16)(21,24,27,38)(22,29,30,44)(23,34,36,49)(25,41,26,39)(28,42,33,47)(31,46,32,37)(35,51,40,50)(43,48,45,52)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=i,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17304);