#129053

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#97
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 170 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)(5,6,8,11,9,7,10,12)(13,14,17,22,19,24,16,21)(15,25,27,36,29,34,26,20)(18,32,31,37,33,23,30,38)(28,42,41,35,43,39,40,44);(2,4)(13,15,18,28,16,27,30,41,19,29,33,43,17,26,31,40)(14,20,23,35,21,25,37,39,24,36,32,44,22,34,38,42);(5,7,8,12,9,6,10,11)(15,26,29,27)(18,33)(20,34,36,25)(23,32)(28,41,43,40)(30,31)(35,39,44,42)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36249);