#498057

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 576 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=fi,d^2=eh,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=eh,g^b=gh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405253);