#108501

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#382
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

522 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=ci,f^b=fh,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15697);