#108409

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	#383
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 566 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(5,6)(7,15,16,20)(8,18,17,12)(9,13)(10,19)(11,14)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(5,7,10,17,11,16,9,8)(6,12,13,15,14,18,19,20)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(2,4)(5,8,9,16,11,17,10,7)(6,12,13,15,14,18,19,20)(23,29)(26,32)(30,35)(33,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,e^a=eh,g^a=gh,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, 15605);