#94280

order	512
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#354
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₂×#391	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#391 ◁ G
LECS	G ▷ C₂×#391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

388 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128

A minimal faithful permutation representation — 2 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,e^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gh,h^b=hi,e^c=ehi,g^e=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1476);