#94351

order	512
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#357
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

168 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 2 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,e^2=fg,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,h^a=hi,c^b=cg

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1547);