#129063

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1770
prime radical	C₂³	#405
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,f^a=fi,c^b=ch,d^b=di,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216227, #72216232, #72216233, …
maximal normal subgroup of	#72216227, #72216232, #72216233, …
maximal quotient of	#72225528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36259);