#153380

order	512
nilpotency class	5
isoclinism	isoclinic class of #153078
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

195 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=di,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fg,g^b=gh,h^b=hi,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60576);