#142296

order	512
nilpotency class	4
isoclinism	isoclinic class of #142236
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#2184	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

650 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|c^2=g,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,g^b=gi,d^c=dh,f^c=fi

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, 49492);