#149336

order	512
nilpotency class	4
isoclinism	isoclinic class of #129554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

526 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dei,c^2=de,d^2=h,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,e^c=eh,f^c=fi,g^c=gi,h^c=hi

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, 56532);