#105819

order	512
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1690 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1690 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=di,f^b=fh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13015);